this video is for those of you who want to understand functions so in this tutorial we are going to talk about relations and functions basics of the functions invest functions composite functions quadratic functions domain and range of the function piecewise functions old and even functions polynomials rational functions modulus functions and radical functions so let's begin our discussion with relation and functions what is elation basically we're not talking about relation we're talking about a set of pairs of input and output values now we're not talking about the input values what are we talking about the input values these are the X values okay at the same time this is the domain so we are saying that this is the pair of the input and output values so we talk of the output values these are y values okay then at the same time this is a range so whenever you hear to say domain we're talking about the X values the length you're talking about the Y values let's say we have got um a function to say a set of Function One comma four then we'll have three comma four let's also have seven comma three okay so as you can see the first part here is our x value this is y this is X this is y this is X this is y now they can ask you to to list the domain The Domain in this case is going to be so we start from from as index is going to be one comma we're going to have three and seven so this is our domain okay what of the range so the range of the values in this case we're going to have uh three we start from three then you're going to have four we only have two so this is our range okay now how can we map this we can see that the X values the ones we are calling them as a domain we're going to put them here we're going to say that this is X and then this is y so to map them we're going to say that we're going to have something like this they're not going to have something like this so we put the values here we can see that we're going to have the X values have got one three and seven the Y values we have got three and four now let's see one is mapping one is mapping Duality four so we're going to have one all the way to four so it's going to come from here or the way to form next we have three to four three to four okay next we have what we have seven to three seven to three now is this a function or not a function so this is a function because all the values of X they are mapping to the values of Y okay so now in short for us to consider a function or a relation to be a function all the input values must have an output value then we're going to call that one as a function not only that but we should also not to say there should be only one output for each input okay so for each input here I need to have only one output perhaps you have seen some equations which maybe we have got this okay is equal to zero it's possible we can find two values of X perhaps we have seen maybe X is equal to plus or minus maybe two meaning of what x is equal to 2 x is equal to negative two okay you can have two values of X mapping to one value of y but it is not possible of you having two values of Y mapping to one value of x so once you see to say one volume uh one value of x is mapping to two values of Y that is going to be wrong that is not a function okay for example let's say we have a function let's have this let's have this let's see if this is going to be a function let's say we have um one comma two then let's have one comma three let's have two comma three let's see if this is a function we are saying that for us to consider something to be a function all the input values must have an output value meaning that all the element of X they are supposed to have the output value at the same time there should be only one output value for each input okay so we have got one we have um we also have two then the output values we have two and three let's see so one is mapping into two at the same time one is also mapping into three then 2 is mapping to 2 is mapping to 3. now we are saying that in we can have two values of X mapping to two different values of Y okay we are saying that we can have two values of X mapping to one value of y that is just okay okay but it is not possible of having two values of Y mapping two uh one value of R of x okay so as we can see here we cannot call this one to be a function because one is having two output it's not possible that we have the input value it's not possible that the input value can have two output values so we are going to say that this is not a function okay so now from there let's also talk about um one-to-one function so let's let's say we have got um a function let's have this let's also have this let's say we have one two three four and sub three four five six so let's say one is mapping to three two is mapping two this is mapping there this is mapping there so we have got X and Y now what you need to understand here is very simple straightforward so this is going to be a function because each input value has got an output value at the same time each output value is only mapping to one um values of Y then we can say that this is one to one this is one two one function because only one element of Y is going to one element or element of X is going to one element of Y one is going to three two is going to four three is going to five four is going to six so this is one to one function let's have another one let's say we have a different one okay let's say we have a zero then we have one we have two and let's have also negative two although I was supposed to start from here this is going to be negative 2. 0 1 then you're going to have we're going to have what we're going to have two just like this then let's have 0 here let's have one let's have four now let's say that this is mapping there then at the same time zero is mapping to zero one is mapping to one this is mapping two so this perhaps this is a function okay let me let me redo it will say this is mapping to one but zero is mapping to zero okay it is possible that two values of X can give us one value of y that is just okay but it is not possible that one value or two values of Y giving us one value of x it's not possible so here we can see that this is going to be a function but this function what type of the function is this this is main to one okay many to one why why are we saying meant one main values of X are going to one value of y as we can see it's negative two and zero they are going to zero only zero meaning that it is many to one so two values of X they are going to one value of y that is it is a function and what type of the function it is mean to one okay then there's also what we call let's have this a different one we see what we're going to have let's say we have a one here let's have nine and let's have a zero okay then this side let's have also a different one so we're going to have um one here here we're going to have three this is going to be negative three zero so let's say that in one is going to one then 9 is going to three at the same time 9 is going to negative three then zero is going to zero remember we say that if we cannot have two values of X of Y going to one value of x so this is not a function one is not a function two this function as we can see it is 1 to main it is not a function but it is one two main function okay why are we saying it is one to many we can see that one element of X is going to uh two uh two values of what two values of Y so it is one to main but it's not a function because we have got one element of X which is mapping to two different uh element of Y which is not possible okay it's possible you can have one two element of X going to one uh one member of Y that is just okay but not one member of X going to two members of what of Y that is not going to be a function okay so now let's say we have a different one like this so we have one let's have two let's have three next let's have this so let's say that this is a this is B so we have four five and then six let's say that one is going to this is going there then this is going there at the same time three is going there at the same time three is also going there so as we can see one element of X is going to uh two elements of Y or one element of a is going to two elements of y t be at the same time two elements of a they are going to one element of what B so this this is not a function it is not a function because one element okay uh two elements are going to uh one element of a is giving us in two different elements of what Y which is not possible okay so now we can say that this is many to many this is men to many but it's not a function okay now let's say you have this so let's say this is X and then this is y let's have n b c and then here we have capital A B C now let's say this is going here okay then this is going there it's also going there so we can see that this is not a function one is not a function because two elements of Y they're giving us one element of x one and at the same time all the elements of X they are supposed to to have an output so this is supposed to be map so if you have got any element which is not mapped to anything as long as that element is in X then that is not a function okay what if I change this then I put it like this okay I put it like this or we say this um goes here then again this one goes there so this is going to be a function it is possible we can have all the element of x which we can we can see that the input has got an output value but still we have got one member which is not mapped as long as that member is in y then that is just okay but we are supposed to have all the element in X they're supposed to be mapped so all the element of the input values they're supposed to have what the output but it is possible that the output cannot have the what the input that is just okay so we can request that one to be a function okay so now let's see how we can solve this we have with this question which is saying find the domain and length of each of the following function so let's start with the function or the domain so the domain in this case we know that these are the X so we're going to start with getting from the ascending order we're going to have zero so we start from 0 from 0 we have three three we have now five five we have got eight eight then we have got nine so this is our domain okay what is the range or the function f the length is one one then we are going to have four four we are going to have seven seven we're going to have eleven eleven we're going to have twelve that is our range okay so go ahead and find the domain and range for the function you can pause the video and the find the domain and range okay so there we go so the domain in this case is going to be we're going to start with some uh negative we're going to start with the the one which is the lowest which is 10. so I've got 10 20 30 40 and 50. that is our domain okay so what is our range in this case the range these are the values of Y so you can see that we have got this this here we have this we have also this so that is our range so we can list the range here we can see that our range is going to be um negative four comma six or is comma one we have got one comma 1 comma six comma 9 so this is our range okay so that is it from question one now the next question is saying which of the following set of ordered pairs represent a function so let's see we have got the first one so the first one we can see that we're going to have so allow me to put X and Y so I'm going to do this okay that is what we're going to have also have these this so what is our X our X we have got one that is the domain I'm getting the the first part here so I'm having this one this one the this one okay so I'm having one three okay I'm having one three and one and seven the uh y values we have what they're four and so we start with three we have got a three and therefore so let's see one is mapped into what four so one goes to four one going to four we put even arrows then three is going to four three is going to form okay seven is going to three seven is going to three so is this a function yes it is a function one because the all the input values have got to add to the output values next we can see that in only one input value is having what only we need to have only one out uh input value for every input value you need to have the output values okay that is the first point then the next point we can see that in it is okay we can have uh two different element of X they can go to one uh to the same element of Y but it is not possible as having two different element of Y going to one element of x okay so this is the function okay so let's let's go to Party B we go to Party B we see if that is a function so we're going to have also X Y we have this I have this what values do we have we have got one we have got two that's all then you have got three and then you have got what we have got two and three here two and three so we can see that in one is mapped into two then one is mapped again into three two is mapped into three so two div one value of x is going to two different values of Y which is not correct so this is not a function so only one value of x is supposed to be going to one value of y t of Y or two elements of X can go to one element of what y that is just okay and not one element of X mapping to two different elements of Y meaning that is not a function okay so let's see we have uh question three here which is saying let's set a to be equal to so we're going to be using that we have been given a and b to be equal to okay we have got the zero one two and three then the B is negative two negative one zero one two now the fifth Point here they are saying which of the following set of ordered pairs represent a function from A to B so the first point which is part A Part A is a one comma zero so one comma zero is going to be we start from one all the way to comma zero so where is one we have got zero comma one so zero here all the way to one next is one and negative two so one here all the way to negative two next is two comma zero two comma zero okay let me redo that one two comma 0 is here next is three comma two three all the way to two so is this a function we can only consider to say it's a function as long as all the values of X all the values of a the appeared they have got an output value okay so as we can see even if this one is not mapped to anything but as long as all the values of X they are they have got the output values then that one is going to be a function then one value of x cannot go to two different values of water two different values of Y so as we can see here all the values here each value has got just one output here so we're going to consider this one to be a function so part one this is we're going to say this is a function okay so let's go to patch B I'm going to use the same one but B is saying we have zero comma negative one zero comma negative one then two comma two two comma two next we have one comma negative one comma negative two one comma negative two that one I'll have three comma zero three comma zero three here comma zero next we have one comma one so one here comma one so this element which is for a this element is giving us two different values of Y meaning that this is not a function so this is not a function because one element of a is giving us two values so one input is giving us here one uh two output which is not possible so we say that in uh there should be only one output for each input as simple as that so this is not a function let's go to the next one we go to part C is part c a function Let's see we have zero comma zero zero comma zero is this point here then you have got one comma zero one comma zero is going there little two comma zero two going there to zero that is just okay then three comma zero again this one going there that is just okay so can if is this a function so this is a function yes because all the output values or all the input values that I've got what output value so we said that there should be only one output uh for each input so we we can see here that this input here has got the output this one has got output this one a good output so this is what this is we're going to call this one as a function okay perhaps you have seen equation which is starting x to the power form you can have four different values of X giving us one value of y so this is a function let's go to Part D so but D is 0 comma zero comma two zero comma two which is there or is zero comma two not negative zero all the way to two one comma watt so we have next is three comma zero three comma zero so my three is here comma zero next I have uh one comma one comma one one comma one that is just okay but all the output outputs all the input values they're supposed to have the output value so as we can see this input here has got nothing we don't it's not mapped to anything so this is not a function okay so this is what we need to know and um relations and functions so let's talk about um uh one-to-one function so how do you know that a function is one to one or is main to one okay so a function is said to be one to one if F of M is equal to F of B okay so if F of a is equal to F of B then we're going to conclude to say this function is one to one so for example let's have um one question here let's say we have a function which is um f of x is equal to x squared minus one so let's see is this function one to one let's prove okay so a function is say to the one to one if F of f is equal to F of B so what it means here is that in in this function where there is X I'm going to put M so I'm going to say it's going to be a squared minus 1 has to be equal to B squared minus 1. so I can shift one to the other side it's going to be plus which is going to be a squared has to be equal to B squared minus 1 plus 1. so these two can cancel out is going to be zero so I'm going to have a squared is equal to B squared so we are trying to make a as a subject of formula what are we going to do is going to be we are going to square both sides now the moment when I Square this side meaning that I'm going to have uh this side this side is going to be plus or minus so it's going to be a is equal to plus or minus B so this this me what it means is that a is equal to b and a is equal to negative B which is not true so we are saying that a function can only be said to be one to one if a is equal to B okay and not a is equal to negative B so as we can see here we've got two values we've got a is equal to b and a is equal to negative B meaning that this function is not one to one okay so this function is not one to one because a is not equal to B so you say after after reaching at this stage you know that a is equal to uh plus or minus B then you're going to say that this function is not one to one since F of a is not equal to F of B as simple as that okay so let's have another example let's say we have a function which is um f of x is equal to X plus C five let's prove let's see if this function is one to one so a function is said to be one to one if F of a is equal to F of B okay so what it means that in the first part where there is X I'm going to put X so it's going to be f plus 5 has to be equal to B plus 5 I can shift 5 to the other side is going to be a is equal to B plus 5 minus 5. so these two guys are going to give us 0. so we're going to have a is equal to B so as we can see a is equal to B meaning that this function is one to one okay so this is how you prove that the function is one to one okay so let's have one more example let's say we have um f of x to be equal to the square root of x let's prove that let's see if this function is one to one so a function is said to be one to one if F of a is equal to F of B so let's see so we're going to say the square root of a has to be equal to the square root of B okay so I can square both side to remove the square root so if I do this I do this I'm going to find that a will be equal to B so this function is one to one okay so another example which you're going to have is uh very simple let's say we have um f of x is equal to 2x minus 3. let's see if this function is one to one so a function is say to one to one if F of M is equal to F of B so as we can see this is going to be 2A has 2 2A minus 3 has to be equal to 2B minus 3. so what I'm going to do I'm going to shift this to the right hand side then I'm going to have 2A is going to be equal to 2B minus 3 plus 3. so these two are going to give me 0. okay therefore I'm going to have two a is equal to 2B so I can divide both sides by two I can see that a is going to be equal to B so therefore this function is one to one since a is equal to B okay um let's have one more example let's say we have f of x is equal to x squared so I believe this is going to be the same as the first example where we're going to say a squared is equal to B squared we get this code both said we're going to have a is equal to plus or minus B so since F of a is not equal to F of B this function is not one to one okay sometimes you might think of um uh relations so let's say we have X and then we have y so let's have this now we see how can you represent a one to one function using mapping so let's say this is um a this is B this is C okay then this is um maybe capital a capital b capital c so if this one goes there this one goes here okay once again this one goes there then this one goes here so as you can see every element of X is only mapping to one element of Y meaning that this function is one to one okay as simple that so using mapping this is how you can tell to say this function is one to one if every element of X is just mapping to one element of Y meaning that is one to one okay so there's also what we call minute one so this is what we call minute one so main to one is whereby you have got a function let's say this is our mapping okay one to see the relationship which is going to be there so let's say this is m okay this is a this is B this is same and then let's have um so if this one is mapping here this is what's going to happen so this is what we call many to one so what is happening here many many are many values of X they are only leading to one value of y so that's why we're calling it as many to one so this is mainly to one function okay now sometimes you it's possible you can use the graph to tell to say this function is one to one okay so if I have my graph here okay then I have got a curve which is like this let's say maybe it's coming from here okay so if you draw a line if this line cuts only one side of the um is only cut to one side of the graph meaning this function is one to one so for example here it's just cutting here meaning this function is going to bond one so for example maybe if you have been given something like this let's say um you have got your graph and then you've got an equation which is like this okay so if I cut my graph here straight I'm going to draw a straight line maybe from here all the way to this side as you can see it is touching physicite okay meaning that this is many to one function so this is not one to one function a one-to-one function is only supposed to touch one side of the graph so if maybe you also have uh something like this this is also not meant one because sometimes you can have this is uh f of x is equal to maybe x squared okay we even proved that this is not one to one so here as you can see it's going to touch two side it's going to touch here and leave here so this is nothing one to one function so this is how you you know that the function is one to one or is not one to one then main to one is when you have got more values of X then they are mapping to one value of y that is mean to one function okay okay so if you've been given a function then they ask you to say find F of 4 find f of negative four find f of negative f of 2. what it means there is that where there is X this is the first equation where there is X we need to put form then we need to find the value okay meaning that what it means there is a f of 4 is going to be equal to in this function where there is X I need to replace it with 24 okay so as we can see there it is it's going to be 4 but X is square so it's going to be 4 squared okay minus six okay then this is going to give us F of 4 is going to be 4 squared is 16. so 16 minus 6 is going to give us a 10 meaning this is the answer okay so we go to party 2 which is saying f of negative 4. so f of negative 4 is going to be negative 4 squared minus 6 so negative 4 squared is going to give us 16. then still minus 6 is going to be 10 so part 1 and part 2 the answer is going to be the same okay let's go to part three part three is saying if we have been given um this function then they ask us to find F of 2. how can we find this so this is going to be 2 squared minus 6. so 2 squared is going to be a 4 minus 6 is going to give us negative 2. so the answer for Party 3 is negative 2. okay sometimes they can ask us to say this function which have been given okay we have been given the function to say f of x to be equal to x squared minus six then they ask us to say given that um f of a is equal to M find the possible values of M so what it means there is that in the first part where there is air where there is x in the first equation we have to replace it with M okay then you have to press it towards to a then you need to find the values of M so what it means there is that we're going to have uh F of a is going to be equal to a squared minus 6. now they want us to put where there is this with a then this is going to be a is going to be equal to a squared minus six so they want us to find the possible values of n so we have our equation now which is the a has to be equal to a squared minus 6. what do what should we do I can just shift uh six to go to the left hand side then eight go to the right hand side so it's going to be six it's going to be equal to a squared minus m okay I can fact out in a now which is the same as I can just write this the same as s squared minus a has to be equal to six I can now fact out M so a will be open brackets then you're going to have a minus 1 is equal to 7 okay now from here we can see we can see that a is going to be equal to 6 and a minus 1 will be equal to six therefore a will be equal to 6 plus 1 meaning that another value of a is going to be 7. so the possible values of a in this case is going to be a is equal to 6 and a is equal to 7 as simple as that okay let's have another function okay let's have different one let's say we have been given F of d to be equal to the square root of 25 minus t squared now they ask us to find they ask us to find F of 3 then they ask us to find again F of um F over X Plus e 5 again they ask us to find F of 2x what can we do so without wasting our time what we're going to do here is what it means here the first part the first question where there is T we are going to replace it with three then we need to find the value okay so there we go it's going to be F of 3 is going to be is going to be equal to F of 3 is going to be equal to the square root of 25 minus where there is T let's put 3 minus 3 squared so F of 3 is going to be 25 minus 9 okay so my f of 3 is going to be 25 minus 9 is going to be a 16. okay 25 minus 9 is going to give us a 16. now the square root of 16 we're going to have a 16 and the square root of 16 is 4. so the answer for part one is going to give us a form okay let's go to party 2 where they want us to find F over X plus five so what would be F of X plus five this is going to be equal to where there is T we are going to put X plus 5 okay so it's going to be the square root of 25 minus Open brackets X Plus e 5 but T is squared so I have to square this so my f of x plus 5 is going to be equal to this is the same as 25 then it's the same as I can say X plus 5 X Plus e 5 and I can I can expand this is going to give me what it's going to give me x times x is going to give me x squared x times 5 is going to give me 5x then again 5 times x is going to give me 5x 5 times 5 is going to give me 25. meaning that this is the part which I'm going to put it there but this is the same as x squared plus 10x plus 2.5 okay because 5 plus 5 is going to give me 10. so I'm going to have minus now this negative is going to affect everything inside there so we're going to have x squared plus 10 x then plus the 25 now we can get rid of this so now from here we can see that our X plus 5 is going to be the square root over we are going to have 25 this is um 25 minus I'm Distributing now negative is going to be minus x squared minus 10 x minus a 25 okay now at 25 and 25 is going to be 25 minus one five so they are going to cancel out it's going to be 0. okay then we are going to have f of x plus 5 is going to be we're going to have negative x squared minus 10x okay so this is the final answer for party term okay now we can also find 33 you can pause the video and find the answer then you continue with it so now there we go we have F of 2x this is uh is going to be where they still need to put 2x so we are going to have a 25 minus so this is going to be 2x but it has to be squared okay so my f of 2 x will be equal to I'm going to have a 25 minus this is going to be 4x squared okay and there is nothing that I can do with this equation now with this function so I'm going to end there so I'll say this is my finance okay so that's what you need to know now let's talk about um the inverse of the function so how do we find the inverse of the function so let's say we have been given a function f of x to be equal to 2 then divided by x minus 2. okay and then let's also have a g of x to be equal to let's just put it as one divided by X plus c one now what if they ask us to find F of invert so in this sometimes they can ask you to say find one over f of x so 1 over f of x e t is the same as F of inverse okay which is the same okay so sometimes they can use 1 over f of x that is the same as the inverse so now in this case we want to find F of inverse now how do we do it to find the inverse of the function what you have to do is you replace this part with y and then make X as a subject of formula okay now after making X as a capital formula where there is y only place it with x then that is the inverse of that function so for example in this case we have f of x is equal to 2 over x minus 2. so where there is f of x I'm going to representative with Y so I'm going to say that y will be equal to 2 divided by x minus 2. my goal here is to make X as a subject or formula so I'm going to say I'll cross multiply here there is one so I'm going to say this is going to be X Y minus 2x this is going to be equal to 2. so my goal is to make X as a subject of formula so I'm going to say that I'm going to shift 2y to the other side x y has to be equal to 2 plus 2y okay from here what I'm going to do I'm going to divide both sides by y both side by y therefore my X will be equal to 2 plus 2y divided by y now what I'm going to do now from here is in this part here where there is y I'm going to replace it with white with X so I'm going to say my f of inverse of X is going to be equal to 2 plus 2x where there is why I'm replacing you back with what x everything divided by what x okay now if you want you can fact out 2 on top there but it's okay you can just leave it here meaning that this is the function of f of x now let's find the the invis now we are saying that our F of this is our F of inverse which is a 2 plus 2X over X let's find the inverse over the function of G so what would be the inverse in this case remember we are replacing this with him we are trying to find this so what I'm going to do I'm going to replace this with Y and the make X as a subject or formula I'm going to say that y will be equal to 1 divided by X plus 1. my goal is to make X as a subject of formula I'm going to cross multip Prime so I'm going to have X Y plus Y is going to be equal to 1 okay I'm going to shift y to the right hand side it's going to be X Y will be equal to 1 minus y next I'm going to divide both sides by y both side by y my X will be equal to 1 minus y divided by y meaning that my f of my G of inverse is going to be equal to now where there is x in this part here where there is y I'm going to replace it with authority with X so it's going to be 1 minus x divided by X so this is my inverse of G okay now from there they can also ask us to find from there they can also ask us to find um they can ask us to find F of G then they say invest so what it means there is that they want us to find F of G first then we need to find the inverse of that F of G okay so this is what we are going to do first I have to find F of G okay so what would be my f of J what it means here is that we did this this is a composite function so what it means here is that in the function of F where there is X I'm going to put G okay so for example here is going to be um my f is 2 over x minus 2. but when there is X I have 2 over X where there is X I'm going to put 1 divided by what X Plus 1 then I have minus 2. now what I'm going to do here I'm going out in Phase 2 uh to finish up with this part the down part here so what would be the down part this down part here it is going to be 1 over X plus one then you have two then this is the same as we have one here so I'm going to say my common denominator is going to be X plus one so X Plus 1 there we're going to have one here minus this one we're going to have a 2X minus 2. so on top there is going to be 2 and the one is going to give us a negative one so we're going to have negative 2x minus 1 over X plus c one this is what we're going to have here meaning we have this now we have 2 divided by negative 2x minus 1 over X plus one so this we have two on top divided by we have a negative 2 minus 1 negative two x minus one then everything divided by X plus one now this again is the same as two times I'm going to get the risk program of this is going to be X plus one everything divided by negative two x minus one okay so now I'm going to say 2 times x and 2 times 1 which is going to be 2X plus 2 everything divided by negative 2X minus 1. meaning that the whole of this is now going to be replaced by 2x plus 1 divided by negative 2x minus 1. okay so I'll continue from here okay I'll continue from here I'm going to say that the whole part there is going to be replaced by we are going to have a 2X plus 2 everything divided by negative two x minus 1. now this this is a f of G now they want us to get the inverse of this but what I have to do here I can fact out negative down there if I want but if if I want to leave it like this I can leave it so this is f of G okay so this is our F4 F of G of X now they want us to find the inverse of this so we're going to replace this part with Y then we make X as a subject of formation so it's going to be Y is going to be equal to 2X plus 2 divided by negative 2x minus 1. okay now from here I'm going to make X as a subject of formula I'm going to cross multiply and I'm going to have negative 2 X Y minus y plus 1 times Y is going to give me y so it's going to be is equal to 2x plus 2. I'll shift 2x to the left hand side at the same time I'm going to shift the Y negative y to go to the other side so what I'm going to have so this is what we're going to have we are going to have a negative 2 X Y minus 2x has to be equal to has to be equal to we we have 2 minus this is it was minus so this is going to be plus y my goal is to make X as a subito formula that is going to be my inverse now so I'm going to say I need to fact out X so I'm going to have X Open brackets I'm going to remain with negative negative 2y minus 2 has to be equal to 2 minus y I'll divide both sides by negative 2y minus 2 even here by negative 2y minus 2. so these they will cancel I'll have my X to be equal to 2X or is 2 minus 2 plus y divided by negative 2 y minus 2. now from here this is my inverse where there is y we are going to replace it with X As We Know so this is going to be now equal to where there is the why I'm going to press it with 40 so it's going to be plus X everything divided by negative 2X minus 2. but if I want here I can I can fact out negative that is going to be very much simple for us what I'm going to do is um I'm going to say um 2 plus X everything divided negative is going to be like this plus C 2. so now I can divide both sides by negative it's going to be negative 2 which is the same as I can start with X going to be negative x minus 2 everything divided by 2 X Plus e 2. so this is going to be my final answer okay so let's talk about a composite function so under composite function you're going to be given um more than one function for example let's say we have our assumption f of x to be equal to 2X squared plus 2 and then we also have G of x is equal to X minus 1 then they also give us to say h of X which is going to be equal to 3x okay now sometimes they can ask us to say find F of G of x what does it mean okay so when you say F of G of X what it means is that in the function of x where there is x in the function of F where there is X we are going to put the whole function of G so it is f of G okay so in the function of F where there is X we are going to put the whole function of G okay so in this case the function of um f is a 2X is this one okay now in this function where there is X we are going to put 20 we are going to put G which is this one okay so we're going to say F of G of X is going to be equal to is going to be 2 then Open brackets where there is X we need to put x minus one so x minus 1 is the function of G okay then since this x is squared I have to square this plus 2. now I need to find the value here although it's not going to be a value but I need just to simplify it okay so I'm going to have X this is the same as x minus 1. at the same time x minus 1. so x times x is going to be x squared x times 1 is going to be negative X then we're going to have negative 1 times x going to have to be negative X then you're going to have 1 times negative 1 times negative 1 is going to be positive 1. so this is going to be x squared minus 2x Plus 1. okay so going to have a 2 Open brackets we're going to have x squared minus 2X Plus 1. okay let's see we have plus e 2 here now we can simplify this is going to be 2 times the x squared is going to be 2 x squared minus this is going to give us now a 4X plus a 2 and then Plus 2. so our final answer is going to be 2x squared minus f of x plus C 4. so this is the final answer F of G of x okay now that is party 1 now let's say maybe they ask us to say find um G of H then they put as X here so what it means there is that in the function of G where there is X we need to put the function we need to put what H okay so what it means is that the function of G is just x minus c one so where there is X you are going to put 3x because 3x is a function of H so it's going to be 3x minus 1. so these I can simplify this so it's going to end there okay now what if I've been given um h of h of f okay I need to find this what it means is that if the function of H where there is X we're going to put in F okay so we are going to have three where there is X now we are going to put FF either 2x squared plus e 2. okay we can simplify this or we can just expand in short it's going to be six 6 x squared plus c r six so we can end here if you want you can leave it there or you can expand leave it there okay now let's have different thing let's have another question okay let's say we have just two functions let's say we have um f of x to be equal to 2 divided by X then you have G of x to be equal to X minus 3. now let's say we want to find F of G of x now like I said where there is z in the function of F where there is X we're going to put G meaning is going to be 2 divided by where there's X we're going to put x minus three meaning I can end there so that is going to be my answer now what if they ask us to find uh G of F that is going to be the opposite now so what it means there is a team in the function of G where there is X we put in F so X we're going to replace it with reality 2 over X is going to be 2 over x minus 3 as simple as that so if you want you can enter there oh you can say that is going to be the same as you have 2 over x minus 3 is going to be the same as if you say x then X there we are going to have 2 minus 3x so you can also end here which is just the same as ending there okay now let's have a different thing let's say we have um again two functions we're going to use F and the G again so let's say we have our F's to be let's say f of x is um 2x Plus 3. then our G of x is um 2 minus x squared okay now if they ask us to find um F over G then of zero what it means that if we find F of G where there is X we're going to replace it with zero and then we find them the value so what we're going to do here is um first we have to find F of G okay so if I'm to find F of G what would be my f of J my f of G of X before I I plug in 0 is going to be in the function of F where there is X I need to put G so in this case we are going to have this is X so it's going to be 2 where there is X I'm going to put 2 minus in x squared plus 3 so this is going to be f o minus 2 x squared plus c 3 I can simplify this and same this is going to be 4 plus 3 is going to be a 7 minus 2 x squared now they are saying that where there is X now we are going to replace it with zero okay meaning that we are going to continue from here we are going to say 7 minus to where there is X we put to 0 0 then we Square it so this is going to give us 2 0 squared is 0 0 times 2 is going to be 0 7 1 0 we're going to get 20 a 7. so our final answer in this case is going to write a 7. as simple as that okay now what if they ask us to find G of f then we say zero okay they ask us to find G of f then we find zero so this is going to be we find face G of f so where is the function of G where there is X we are going to put in F G is 2 minus x squared now where there is z that is f now so it's going to be 2 okay minus I'm going now to put this in Brackets 2x plus 3 then I have to square it so this is going to be 2 minus this is the same as 2 times 2 is going to is 2x times 2x going to be 4 x squared then 2 x times 3 is going to be Plus in 6X again 3 times 2x is going to be plusing 6 X again then 3 times 3 is going to give us what is going to give us a 9. okay so this is going to be 2 minus this is going to be a 4x squared plus a 12 X plus C 9 I can now distribute the negative so it's going to be 2 minus 4 x squared minus 12x minus 9. so 2 and 9 can go so it's going to be 2 minus 9 is going to be negative 7. so I'm going to have negative 7 minus for x squared minus um 12x now they are saying that well there is X we put 0 so this is going to be negative 7 minus 4 0 squared minus then we're going to have 12 again 0. so this part is going to give us 0. this part is going to give us 0. so our final answer will end up having negative seven so that is going to be our answer now Kim now we can we can have a different one where we say the same one let's let's try to find F of f then we need to find four so what it means that in F of f in the function of F word is x we put F again so our F in this case is a 2x now where this x is 2x plus 3. so whether this x we're going to put the same function at this is going to be two Open brackets then 2x plus c three then again plus 1 3. okay so it's going to be 2 so we can say 2 times 2 is going to give us what it's going to give us a 4X plus this is going to be a 6 then Plus in three okay so now uh we can see that in from here is going to be a 4X plus 9. where there is X we put four so it's going to be four times four plus nine four times four is nine um for the M4 is 16. when 16 plus 9 is going to give us a 25 so 25 is the answer in this case so let's talk about how to find the domain and the range of the function so whenever we're talking about the lens and the domain so let's start with the domain whenever we're talking about the domain these are the values of x okay and whenever we're talking about the lens these are the values of Y so let's say we have the function f of x is equal to 2x Plus 1. now whenever they ask you to say find the domain on the function the values of X which when we plugged in here the function is going to be defined so in this function I can plug in any value of x this function still is going to be defined any value meaning that the domain is all real numbers okay but now let's say you have a function uh let's say you have a function which is f of x and then you have maybe one divided by X plus one then they ask you to find the domain now here you can say it is all real numbers because we have the fraction so on the denominator here once we plug in 1 or let's say once we plug in negative one this is going to be 1 over negative one plus one which is going to be one over zero this is going to be undefined but we are trying to make the function to be defined okay so in this case the domain is not going to be audio numbers now we never have been given a fraction what you have to do get the the denominator set it not equal to zero so you say X plus 1 not equal to zero why are we saying not equal to zero because we can never have a zero on the denominator once we have zero on the denominator the function is going to be undefined that is very very important you have to know it the moment I put 0 if I'm going to have one over zero or maybe I have two over zero this is undefined and we are trying to make the function to be defined that is the reason why we have to set the whole denominator not equal to zero okay then solve for x the value of x which you're going to find that is going to be the domain so in this case it's going to be X not equal to 0 minus 1. X not equal to negative one so in this case the domain is going to be X is such that X is not supposed to be equal to negative one but X can be any real number so any real number I can plug in any value of x any value as long as that value is not negative one if this function is going to be defined okay now you can represent domain either in set interval notation or insert Builder notation so this is the set to the notation which you have if you want to represent it into um set interval notation you can say that the domain is going to be from negative Infinity from negative Infinity blank light negative Infinity here okay all the way to negative one but negative one is not included Union from negative one also not included all the way to positive Infinity so this is going to be my domain but negative 1 is not included that's the reason why I'm using open okay the moment I use cross meaning it is included so it's not included now how do we find the length the range these are the values of Y now to find the length what you have to do is what you have to do is you need to find the inverse of this function once you find the inverse of this function the same method which we are using on how to find the domain that is going to be the same method which you're going to use to find the range so range these are the values of Y so when you're finding the inverse of the function we put this we'll present it with what y then make X as a subject of formula so it's going to be with cross multiply is going to be Y X or let's say x y is going to be X Y plus Y is equal to 1. my goal is to make X as a subjective formula this is going to be X Y it's going to be equal to 1 minus y I'll divide both sides by y plus side by y x will be equal to 1 minus 1 over 1. now the values of whenever we're talking about the range those are the values of y t the the values of Y so we know that the inverse is going to be the inverse of this function well is X we are going to put it's going to be 1 minus X where there's y represent it with what x then we have this this is the inverse of the function now the same method which we use to find the domain that is the same method which you're going to use to find the range so I'm going to get this the denominator I set it not equal to zero so that would be my range so lens is going to be Y is such that y should not be equal to 0 y can be anything but should not be equal to zero at the same time y can be the number of oriole numbers so that is my range so to find the range just find the inverse that would be the range what next let's say we have another question which is um f of x f of x is equal to s of x plus 4 over 2x plus c let me put minus minus 1. let's find the range on the domain you can pause the video and work on it okay so to find the domain we need just to get the denominator set it not equal to zero and find the value of x 2x is going to be not equal to zero then you are going to have 0 plus 1. so I'm going to have two x one so we divide both sides by 2 both side by two x not equal to one over two that is our domain so the domain will be equal to X is such that X should not be equal to 1 over 2 but X should be the member of all your numbers so this is the domain you can put this in set interval notation where you can say the domain of B from negative Infinity to half note included Union half note encoded comma positive Infinity so that is the domain okay what else let's now see how we can find the lens we need to find the inverse of the function the same method which we used on how to find the domain that would be the same method which you're going to use to find the range so why would be equal to X Plus 4. to x minus 1. so we cross multiply we are going to get um 2xy minus y is going to be equal to x minus plus 4. our guys to find the X to make X as a subject of formula we can shift this x to the other side and the one to the Lesser so we're going to have 2xy minus X is equal to 4 plus one which is going to be 5 that side so you have 2 X Y minus X is equal to 5. we can make X as a subator formula this is going to give us 2y minus 1 is equal to 5. we divide both sides by 2 y minus 1 both sides by 2y minus 1 X will be equal to 5 over 2y minus 1. so that is the inverse of the function so this inverse here if they ask you to find the inverse you just say the inverse is going to be negative 1 f of x is going to be 5. divided by 2 where there is why you put X so this is the inverse now you get this because we're trying to find the values of Y so you get this 32 not equal to zero so it's going to be 2y minus 1 not equal to zero so for y okay I believe this is going to give us the same as the domain so it's going to be 2y is going to be not equal to 1. so I divide it by 2 we're going to get this we have uh 2y not equal to 1 we divide it by 2 we divided by 2 y not equal to half so the length in this case is going to be y a range is represented by y because we're talking about the values of y y is such that y should not be equal to half but why should be the member of oriole numbers you can also put this inside interval notation where you say it's going to be negative infinity or the way to half then Union half or the way to negative Infinity positive Infinity so that is the range okay so let's have another example where we have something which is complicated just a bit what I'm going to have let's say we have um f of x as um 3x let's say 3x plus 2 over X squared minus 4. get the denominator set them not equal to zero x squared minus 4 not equal to 0. x squared not equal to 4. so I get the square root both sides I get the square root meaning here I'm going to have plus or minus so how X not equal to plus or minus 2. meaning that in my domain is going to be X is such that X should not be equal to plus or minus 2 but this x has to be the member of audio numbers so I can also put this insert interval notation where I can same um where I can say I need to have the values the domain is going to be from negative Infinity all the way to negative 2 not included Union then two all the way to Infinity but 2 and negative 2 they're not included okay now how do we get to find the length find the inverse of the function the same method which we use on how to find the length that would be the same method we are going to use now you can even test here we are saying that X should not be equal to plus or minus 2 and for sure if I plug in 2 here it's going to give me 4 which is going to 4 minus 4 0. on the denominator there I'm going to have zero which is going to be undefined probably want our function to be defined so what are we supposed to do 2 is not supposed to be included negative 2 is not supposed to be included because once I plug in negative 2 then I'm going to get a form four minus 4 0. so it's not supposed to be included so that is making sense let's find the inverse now so to find the inverse we say y will be equal to 3x Plus 2. 3x minus 4. this is going to give us three x squared y minus 4 is equal to 3 X Plus 2. so what I'm going to do is um I'm going to shift x 3X to the left hand side and negative 4 to the right hand side so I'm going to hold on minus 3x is going to be equal to 2 Plus 4. so we're going to have 3 x squared y what else I'm going to have we're going to have uh we're going to have what we are going to have a 6 here okay oh I forgotten minus 3. then you have 2 this is going to give us six so this is X we are trying to make X as a subject or formula what do we do we fact out X okay so if I fact out x squared although this is not going to make sense what I'm supposed to do here is okay so this is going to be um we're going to have what we're going to have 3 y minus so here I'm going to the menu with again 3 over X which is not making sense at the same time okay so if it's not making sense the meaning that this function has got no inverse it has no invest so it has got no inverse okay so you leave it there let's say we have another question which is um so if the function is going to invest it has got no range because you can't find the range there so it has good no but it has got no uh the values of working the values of Y so it is not restricted actually so we have the next question which is uh let's say we have now the radical one f of x is equal to this okay how do we find the domain we know that we can never have a negative let's say we have a negative we can never have a negative inside the square root that is going to be undefined okay meaning that what we have to do is we need just to get what is inside them greater than or equal to zero we can we can have zero we can have zero we have the square root of zero which is zero zero going up or let me say zero and above that's making sense the function is going to be defined but we can never have a negative so in this case the domain is um X is such that X has to be greater than or equal to zero at the same time x has to be the member of warrior number using the set interval notation you can say that the domain will be starting from zero but zero is included so I'm supposed to use the closed one starting from zero or the way to positive Infinity but we don't know if possibility is included so put the open one so this is the domain now to find the length you need to find the inverse of this function finding the inverse you know what to do we are supposed to put X Y is equal to the root of x what else we Square both sides to remove the square root because we're trying to make X as a Sub 0 formula we're going to have y squared is equal to X okay so what I'm going to have from here we can see that is the same as we have a function which is X um we have X is equal to Y to the power 2. so we have nothing which is involved in um this part here I can plug in any number the function is going to be defined okay yeah the function is going to be defined so what values of Y if I plug in there the function is going to be defined okay so this is going to give me audio numbers this is going to give me audio numbers now we are going to talk about graphing of radical functions so also talk about how to find the domain and the range of radical functions okay so let's start let's say we have a function which is um f of x is equal to the square root of x now last time we mentioned to say whenever you are trying to come up with a domain you have to set what is inside greater than greater than or equal to zero okay so I have to say x is greater than or equal to zero meaning that that is my domain but you can also uh find the domain and the lens especially the range when it comes for the lens for the radical function it's very easy for you to not to know the range when you sketch the graph okay so here we are saying that X has to be greater than or equal to zero meaning that the values of X which are which you are supposed to use they're supposed to be greater than or equal to zero that's when this function can be defined so we have X Y so I'm going to say that in well I'm going to start from 0 0 is included so if I put 0 where there's X the value of y is going to be 0. next if I put one the value of y is going to be 1 because the square root of 1 is 1. so what if I put for the value of y is going to be 2 because the square root of 4 is 2. next I can say nine nine the square root of nine is three I can end that if the one you can have three coordinate that's just okay so next what we're going to do is um we're going to have our graph okay here is going to bar graph why this is our X so we're going to have one two three four five six seven let me put another one here and another one so meaning we have one two three four five six seven eight nine okay so we have um in the y axis we can have one two three four that is just okay so next we can plot we have zero comma zero is starting from this point then one comma one is this point we have a four comma two so this is a two so three four four is this one comma two is somewhere here okay what next we have nine comma three three is here nine is somewhere here so it's supposed to be somewhere uh here so our graph will start from here it goes there start from there it goes there so that is the graph of the square root of x now from here you can find the domain domain these are the values of x so what the what values of X do we have as we can see from the graph here we have go to zero it's starting from zero going to positive Infinity meaning that our domain is a 0 is included or the way to positive Infinity that is the domain now the length range these are the values of y 0 is included so it's also starting from zero all the way to positive Infinity okay so that is the parent graph which we have for The Logical functions let's have a different thing now let's say we have um f of x to be equal to and the square root over 2 minus X so what you have to do for you to know which one so let's talk about a piecewise function so what is a piecewave function so a piecewise function this is the function she has got more than one equation perhaps you have seen an equation which is like this or a function which is like this Kim so you're going to have um maybe more than one function let's say we have got X plus 2 then the domain is um X is less than or equal to negative one so each function represents has got its own domain okay so we have one then you have got comma negative one is less than x X is a less than one then we also have x squared then we have common X is greater than or equal to one okay now here they're going to ask you to sketch this and maybe find the value sometimes let's say they ask us to say find um F over f of negative one also find F of zero again find F of three so how do we find that so before we go any further let's face sketch the function okay so sketching a piece y function it is very very simple what you have to understand is that you only follow the domain which have been given so you only follow these These are the uh the domain okay so the first part here this is the domain for the first function this is the domain for the second function this is the domain for the set function okay so now what is happening here is that you what we're going to do we're going to say this is our our graph so I'll put it here this is going to be my graph okay so here's my graph now this graph what I'm going to do is I'll start with the coordinate which I've been given so as we can see from the domain we should we have been given we are going to be following the same the same conditions which have been given so the first part they're saying that X plus 2 the values of X are supposed to be less than or equal to 1. okay so I'm going to start there I'm going to say this is going to be my X this is going to be my y so the first part is negative one so if in this function in this equation if I replace negative 1 where there is X what will be the value of y okay because the the condition here is saying that X is less than or equal to negative 1. so I need to start from negative one but at that point since it is less than or equal to the first point is going to be a closed one meaning it is included if it was X less than 1 or negative 1 I would have started with what an open one meaning it is not included okay so here I'm going to start with that one then I'm going to say if I put negative 1 in my first equation I'm going to find that it's going to be negative 1 plus 2. so what is negative 1 plus 2 I'm going to get a one okay so I'm going to get a 1 then if I put negative 2 is going to be negative negative 2 plus 2 is going to give me a zero so I'm going just to get three um three numbers or three values of X then let's also put negative three if I put negative 3 I'm going to find negative one now from here what I'm going to do is um I'll go to My Graph okay the X values which I have is a I have negative one negative 2 negative 3 allow me to put also negative form so I have these values here next I'll go to my y my y I have got negative 1 negative two I'm going to put all the way up to negative three that's all okay then again the first part here I'm going to put okay here I'm going to end on just one for the first equation so I'm going to have negative 1 negative 2 negative 13 here I've got one so let's now plot the first um we should have so what is negative one negative one comma 1 is this point here okay but at this point we are saying that it is included because it is X is less than or equal to negative one so negative one comma 1 is included so I'm going to put an a closed one so one common one is going to be at this point I'm going to put a closed one meaning it is included okay I've put that one next I'm going to to go well is negative 2 comma zero so where is negative two comma zero negative two comma zero is this point here okay the negative three comma one negative three is this one one is here so it's somewhere here meaning that my point here is going to start from this point or they were going here that is my point so that is the first equation which you're going to have meaning for the first part I'm done for the first equation I'm done my sketching I'll go to the second one okay so what am I going to do in the second one the second one what I'm going to do is I'm going to have also X and Y but this one is is saying the values of X are supposed to start from negative 1 all the way to one meaning that I've got a negative 1 here negative 1 0 and the one that's all that is the domain so the domain for that one is just from negative one negative one not included all the way from from negative one not included to one so this is the domain which I've been given okay so that is what we're going to use those are the values of X so in this one if I put negative 1 I'm going to find that my value of y is going to be one I've been given already here so any value which is within the range of negative one all the way to one is going to give me one so if I put negative 1 it's going to give me one but T is not going to be included at that point 0 is going to be giving me one one is going to give me one so meaning that the first point is not going to be included and which point is that negative one comma one a negative one common one is this point so I'm going to put an open circle here to show that it is not included next I'm going to say 0 comma 1 0 command is this point okay next is going to be one comma one so one I'm going to put one here okay one comma one is going to be at this point it's going to be at this point meaning that at this last part it is not or it is less than as well so it is going to be an open one it's not included as well meaning that I need to put now the line joining these two points I'm done right so the second equation that is my graph that is my graph okay now from there I'll go to the third one now okay how do I go how do I sketch that one so I'm going to say that I'm going to have this X Y so let's start now putting the numbers so I have the thing that X is greater than or equal to one one is included so I'm going to if I put one one squared well this x I put one one squared is going to give me one let's put 2 2 squared is going to give us what 2 squared is going to give us some F4 okay let's put three okay same squared is going to give us what is going to give us a nine so what we're going to do this this is one let's put here two that we put here for then here is put name because we don't have space so I'm going to say that this is going to be my four this is going to be my 9 meaning that from here now I can say one comma one but at this point one comma 1 is going to be included so I'm going to put it inside there okay so allow me to say this point here let me just say since it was open I'm going to put an open one there okay then now since this one is going to be closed I'm going to put a closed one starting from there that is where it's starting from now next point is 2 comma four so two comma 4 I'm going to put two here and then I'm going to put three here so 2 comma 4 is going to be this point okay this point here is going to be 2 comma four next is going to be 3 comma 9 3 comma 9 is going to be somewhere here okay this point so I need just to join these lines so I joined these lines that is going to be my third one so now this graph which I have is the piecewise function which I have this is the graph now okay now from this graph we can see that we can even find the domain the overall domain of this function we can see that the X values we don't know where it is ending so it's going to be all real numbers starting from negative Infinity all the way to positive Infinity that is going to be our domain because we don't know where it is ending that's why I'm putting the arrow these arrows simply means that we don't know where it is ending okay so it can go all the way to positive Infinity that is X all the way again to negative Infinity so it's going to be uh the the domain is going to be all real numbers what of the range the Y values if we can check here the Y values are not restricted as well so the Y values as we can see is starting from negative Infinity as well all the way to positive Infinity as simple as that okay now the next question you should have to ask ourselves is what if we want to find f of negative one okay how are we going to find that so f of negative 1 this is what we're going to have okay so F over negative one so from these equations which we have here we're going to choose the one which is telling us to say x is equal to what negative one so the first equation saying that X is less than or equal to negative 1 meaning that negative 1 is included I'm going to get that equation so I'm going to say that X plus 2 okay so f of negative 1 is going to be negative 1 plus e 2 so what will be our f of negative one f of negative 1 is going to be negative 1 plus e 2 is going to give me 1. so that is there answer for what for for party one okay cool now let's let's also find the second one the second one is saying find F of zero so we're going to choose the equation which is telling which is going to tell us to say x is what zero so what which equation is that so we have got F of zero the first equation saying that X is less than or equal to negative one meaning that the numbers from negative one all the way to negative Infinity 0 is not included there we can't we can't get the uh equation one let's go to equation two okay equation 2 is saying that negative 1 or the way to one but those two are not included but in between there is zero meaning that that is the function which you're going to get okay meaning that F of 0 is going to give us what one because we don't have anything apart from one here so F of 0 is going to be um is going to give us some one that's all now the third one is saying F of three how can we find F of 3 okay so F of 3 this is what we're going to have F of 3 is going to be uh which equation you're going to get we can't get the the first equation because the first equation saying that X plus 2 then X has to be less than or equal to negative 1 so we can get that one the second equation is saying negative one or they were to one series is not there but the third equation is saying that X is greater than or equal to negative one meaning that the numbers from negative one or the way to positive Infinity so 3 is going to be in that one meaning that we are going to say this is going to be F of 3 is going to be 3 squared so F of 3 is going to be 3 squared is going to give us 9. so that is it for that one okay so now let's say um we have a different we have got a different function let's see how we can solve it okay let's see how we can solve it so let's say we have got a different one and the same different one let's say um f of x is equal to okay so I will draw this um this is zero that is um then we start from negative five all the way to negative 5 less than or equal to X then X has to be less than a negative 2. okay now when I say negative 5 less than or equal to X then less than negative 2 I can put this insert interval notation so what it means there is that negative 5 is included comma negative 2 which is not included so what it means is same as X is greater than or equal to negative 5 at the same time x is less than negative two meaning that the starting from negative 5 negative 5 is included or they were towards this but this is not included that is what it means so now what we're going to do here is a um the second one let's say we have um Negative X squared plus four comma let's put negative 2 less than or equal to X then let's say x is less than or equal to negative one so let's let's include both of them so let's say also here we have Negative X then we have plus three comma let's say a two is less than x then X is less than or equal to 5. now let's sketch this we have to sketch and then we need to find um one we need to find f of negative one again we are going to use the same one F over we need to find F of zero then this third part we have to find F of three so first let's sketch the graph okay so sketching this one is not going to be complicated because um we know that we're going to have this okay that is what we are going to have let me have a nice one so here is my graph so what I'm going to do is I'll start with the first part I'll start with the first part so the first part we know that we are going to have X Y so we have this now they are saying that the values are supposed to start from negative five all the way to negative two so I'll start with negative 5. if I put negative 5 the Y value is going to be 0 next is going to be negative 4 negative 4 the Y value is going to be zero because we don't have any x value here meaning that that is the answer if I put negative 5 that is going to be 0 I put negative 4 this is going to be 0. I put negative 3 this is going to be zero all the way up to negative two if I put negative 2 this is going to be 0 okay so now from there what I'm going to do is I'm going to start now putting this I'm going to have my negative 1 negative 2 negative 3 negative 4 negative 5 negative six so I'm going to put here negative 1 negative 2 negative 3 negative 4 negative 5 negative six so they are saying that it is starting from negative negative 5 comma zero but negative five is increase so I'm going to put a closed one okay it is included because it is less than or equal to so negative 5 is included next negative 4 comma 0 is this point negative 3 comma 0 is this point negative 2 comma zero it is also included no it is less than two so it's not included I'm going to put an open one there so my line is going to start from here all the way to this point so that is going to be my first part okay now next I'll go to the second part so I'm going to have this as well X Y now let's put the values they're saying that from negative one all the way to one so I've got negative one sorry negative two all the way to one so negative two we have got negative one zero all the way to one so in this equation is x negative x squared plus 4. so if I put negative 2 now remember when you have something like this this is this what when you're plugging in the values what it means is that it's going to be negative then negative 2 squared plus E4 so this is going to give us inside there's going to be four but for this negative outside is going to be negative four so negative 4 plus 0 is going to give us a zero that is also very very important don't make a mistake where you are going to say you plug in 2 then you think as this is going to be positive no because negative is outside the X okay so we're going to have another one which is going to be um is going to be negative x squared plus 4. so if I put negative then I have negative 1 I square is plus 4 it's going to be 1 squared is going to be negative 1 squared is going to be one so there's negative out there so it's negative 1 plus C 4 which is going to body it's going to give me uh three okay so I'm going to put my 3 here what if I put 0 if I put 0 I'm expecting to have a 4 because it's going to be 0 squared this is zero zero plus four we have four next we'll have one okay so the first point is going to be included it is included even this the last point is going to be encoded so the last point is going to be one if I put one there it's going to be one square so it's going to be negative one comma I'm going to have um I'm going to have what if I have a 1 squared is going to be it's going to give me S3 okay it's going to give me a three so I'm going to start now from there to say the first point is included so I'll come there and say my first point I'm going to have this dish is included which is negative 2 comma 0 is this point so I'm going to put it inside it is included yes okay it is included next negative one comma three allow me to put uh one here let me put two I put three here let me put four I put five so I have one two three four five so next point is um the first point was a negative two comma 1 that is the first negative two comma zero that is the first point which I've shaded it then next is negative one comma three is this point here then we have zero comma four okay which is this point next we have uh one comma three so I'll put one here three easy here so it is starting from here okay let me use different column it is starting from this point or they were there then here but the last point is also included so I need to put a I need to share it so that is the second part now the third part what I'm going to do is I'm going to say E I have X again I have y so I have this so I'm going to have a setting from 2 2 is not included so the first point is going to be open okay so 2 is not included if I put 2 there what's going to be my answer negative 2 plus 3 I'm going to have a 1. okay I'm going to have a one next I'm going to put the S3 is supposed to end one five three so if I put 3 there negative 3 plus 3 is going to give me a zero next I'm talking about this equation here that's why I'm plugging in the values so now um next I'm going to have what I'm going to have a 4. if I plug in 4 here negative 4 plus C this one I'm going to have a negative one okay next I'm going to have a five but at this point where this 5 is going to be included because it is less than or equal to so if I put 5 it's going to be negative 5 plus 3 so what is negative 5 plus 3 negative 5 plus 3 is going to give me a negative two so I'm done now from there the X values are supposed to go all the way up to five so I've got one two three four five let me input six so I have got one two three four five six as simple as that next what I'm going to do is I'm going out to plot this so allow me to put negative 1 here then I put negative 2 negative 1 negative two so I'm going to have um two command okay so 2 comma 1 is this point but at this point it is open it is nothing is going to be open next is Phil comma 0 is this point three comma zero is this point next four comma negative one four is this point negative one is somewhere here okay another one is five comma negative two five is here negative two is somewhere here so our point will be here which is going to be our point okay so don't forget to put the arrows because we don't know where it is ending so this is the sketching will be our f o we start with f of negative one okay so f of negative 1 which function are we going to get let's start with the first function so the first function saying negative five um negative five is less than or equal to X then X is less than negative 2. negative 1 is not in that range so the first equation has failed let's go to the second one negative two less than or equal to X then X is less than or equal to what to 1 meaning that that one is going to work because it is from negative 2 all the way to one so negative one is in between so we're going to get that function it's going to be negative x squared plus 4. now we plug in the values negative one so we're going to have negative negative 1 squared plus E4 so what will be our f of negative one so f of negative 1 is going to be negative 1 squared is going to give us a one since we're going to have negative one now plus 4 because of the negative which is outside so f of negative 1 we're going to have a three so this is going to be the answer for f of negative 1. okay now what will F of zero which one are we going to get same so F of um f of zero let let's check the first one so the first one is saying uh from negative five all the way to negative two zero is not in that range this the second one is saying from negative two all the way to positive one meaning that zero is in that range so we're going to get that one again so it's going to be negative x squared plus 4. so F of zero is going to be then we have 0 squared plus four so our answer is going to be F of 0 is going to be 0 plus 4 is going to give us F1 so that would be the answer for f of 0. okay so that is also very very important you knowing to say which equation am I going to use if I've been given F of 2 F of what yeah so F of 3 now which one do we expect we expect the third one the third one and that's why saying two or the way to five meaning that three is in that lane so it's going to be I'm going to get this equation okay so F of 3 is going to be negative I'm going to replace where this x will pass through 33 this plus three so F of 3 is going to be equal to zero as simple that so this is what you need to know and uh piecewise function so let's talk about uh even and odd functions so how do you know that a function is um even or odd so a function is said to be an uh to be even if f of negative X is equal to f of x meaning a function is what even and the function is said to be odd function if f of x if f of negative X is equal to negative f of x okay then we're going to say that this easy and if an odd function so if you discover to say the function is not giving us this and at the same time it's not giving us this meaning that function is neither of the two is neither even also just say the function is neither okay so there we go let's have some examples okay so let's say we have got a function which is a f of x is equal to 3x squared minus 4 x to the power 4. let's let's see if this function is odd even or neither okay so there we go so a function like we said a function is said to be even if f of negative X is equal to f of x so if we plug in negative well there's X and we get the original function then that function is even but if we plug in the value of x with negative X and we get negative f of x then the function is odd okay so for example we have got an example here so this is going to be f of negative X is going to be equal to so it's going to be 3 negative x squared minus 4 Negative X to the power 4. okay so we are saying that in this is going to be f of negative X is going to be equal to negative x squared is going to give us x squared so it's going to be 3x squared minus um x to the power 4 is going to be x to the power 4 so you're going to have this so as you can see we are getting f of negative X then we are getting the original function which is going to be equal to f of x so since f of negative X is equal to f of x then this is an even function okay so this is how you prove to say it is an odd or even now let's have another example let's say we have F of X is equal to 3x squared minus 4 okay this is what we're from having let's have this f of x is equal to um x to the power 3 plus 1 okay so we know that for a function to be uneven and even f of negative X has to be equal to f of x and for a function to be an odd f of negative X has to be equal to negative f of x so let's see if this function is even or odd function okay so we're going to say f of negative X is equal to we're going to have negative x to the power three plus one so f of negative X is going to be equal to Negative X Plus e 1 so let's try to factor out a negative okay so f of negative X is going to be equal to Negative X then again we're going to have negative here negative one so as we can see um f of x negative X is not equal to make this f of x so this function is not even what of want okay or unless if we have got a negative outside then inside here it's supposed to give us the original function but at the same time this function f of negative X is not equal to negative f of x so this function is not an even at the same time it's not an odd so this function is neither as simple as that okay so let's have another example let's say we have um a function which is um F over um f of x is equal to 6 x to the power 3 minus 5x so let's see we are saying that for a function to be uneven f of x f of negative X has to be equal to f of x and if a function to be an odd um f o x has to be equal to negative f of x okay so let's now see what if what's going to be our f of negative X so f of negative X you're going to have 6 x to the power 3 minus 5 Negative X here okay so what I'm going to have F over Negative X we are going to have um Negative X to the power 3 is going to give us negative X is going to be negative 6 x to the power to the power 3 the minus what about negative negative is going to give us a positive 5X so let's Factor the negative f of negative X is going to be equal to negative then we're going to have X to the power 3 minus 5X so as we can see we have got a negative here so we've got f of negative X is equal to negative f of x now inside here we need to get the function which is exactly the same as the original function if we factor out negative then we're going to conclude that this since this is negative f of negative X is equal to negative f of x then this is an odd function okay let's have another example so let's say we have um f of x is equal to 1 minus x squared so let's see if this function is even or odd so let's not forget for a function to be an even f of negative X has to be equal to f of x if a function to be an odd f of negative X has to be equal to negative f over X as simple as that so let's see so we have got F over Negative X is going to be 1 minus negative x squared so we're going to have f of negative X is going to be equal to 1 minus negative x squared is going to give us x squared so this is giving us exactly the original function so since f of negative X is equal to f of x then we can conclude to say this is an even function simple that okay so let's have another example let's have another question f of x is equal to x squared minus X let's see so f of negative X is going to be equal to um negative x squared minus negative X so this is going to give us f of x is equal to um next x squared we're going to have x times negative x times negative X are going to have this okay so f of negative X is not equal to f of x so this is not uneven at the same time f of negative X is not equal to negative f of is not equal to negative f of x so it's not an odd not odd therefore this function is neither so this function is neither is neither old um no even okay let's have another example so let's say we have um f of x is equal to three x squared let me put yeah let me put x squared then plus 2X minus 1. so let's see so function to be an odd f of negative X has to be equal to F over X let's not forget that point very very important at the same time for function to be um to be an odd function f of negative X has to be equal to negative f of x so let's see so f of negative X is going to be 3 negative x squared plus 2 negative x minus 1. okay so f of negative X is going to be equal to 3x squared because negative x squared is going to be x squared minus this is 2 times negative X going to be 2x minus 1. okay now um this is not giving us exactly the same as the original function okay so the original function is this there's a plus here and here there is a minus so it's not the same now let's try so since it's not the same it's not an even let's see if it's an odd let's factor out the negative so if I factor the negative I'm going to have negative 3x squared plus 2X minus 1 or plus one but again I'm having some changes here negative and then here which is not the same so this again is not the same as the original one so this function is an odd function therefore is neither so if you discover to say it's not an old and it's not an even then the function is neither okay so now let's have another one let's say we have f of x we have f of x is equal to X Plus 1 over X is this function even or odd okay so f of negative X is going to be Negative X Plus 1 over Negative X again so this is going to be negative X is going to be equal to we're going to have Negative X this negative they are going to have negative 1 over X now this is not the exact as what we have there therefore is nothing an even function let's try to factor out the negative okay so if I try to factor out the negative I'm going to find that I'm going to have Negative X here plus 1 over X now we can see that what we have inside here it says same as the original function therefore since f of negative X is equal to negative f of x then this function is odd as simple as that so this is how you know to say this function is odd or even okay now let's have the last uh example let's say we have f of x is equal to the modulus of X okay now since we know that any value which is going to come out from here is going to be positive now we are saying that if if this is giving us this then this is an even if this is giving us negative f of x then this is an odd function so now let's see if this is going to be even odd or neither so we have got f of negative X is going to be equal to Negative X now we know that in modulus we're going to come up with only positive values so this is going to give us a positive X okay now if this is positive X we can see that this has given us the word to the original function okay therefore this is the and um an even function as simple as that so this is how you know or this is how you determine to say the function is an odd even or neither okay we are going to talk about Computing Square methods okay so Computing Square method we are going to have a function which is in form of f of x is equal to a x squared plus BX plus c now the first step here is we have to factor out m okay so if I fact out a I'm going to remain with a x squared plus b over f x plus C over a okay so the first point here is you always fact out in a the reason why we are factoring out a we want our our turning point to be in a form of a Open brackets X Plus p okay then we have squared plus Q now that's the reason why we are factoring out working the air okay so we will keep on bringing the air there so now the next step here is you get the coefficient of this x which is here now you do times half okay what is the half of B over s so you say B over a times half which is going to be B over 2m okay so this is going to be a x squared plus b over a X Plus now this is going to be B over 2A now this has to be squared the same thing which I've done here I'm going to say minus the same thing now B over 2A I also have to be square the reason why I'm doing this is if I do this minus this I'll go back to my original equation okay then I'm going to to put now this so this is going to be plus now uh I have C over M then I do this so what I'm going to do here is I'm going to ignore this so now if the moment I ignore this I know that this is squared this is also squared then it can be written in this form it can be written in a I have X plus b over 2A then I do this I put a squared I have not changed anything so I'm going to say minus this one I can say B times B is going to give us some a b squared then this is going to give us a form a squared this is going to be plus C over m so I have this now from here what I'm going to do now is I need to to make this as one thing what I mean there is a I can get rid of this on top here I can say that I have negative B squared over 4A squared plus C over m the lowest common denominator is going to be 4 a squared so this there is going to be one one time that one you're going to get a negative B squared plus a here we're going to get a 4M for a time that one we're going to get a form SC meaning that I'm supposed to replace this with the what I have there so I'm going to have a Open brackets I have this okay I Square it then I have minus B squared plus I have four SC everything divided by what 4A squared and I have this so what I'm going to do I'm going to distribute now m so if I distribute a you're going to see that this is what we are going to have let me just get rid of this so what I'm going to have here is um we're going to have m X Plus B over 2m infinity squared so minus if I put this this is going to be canceled with one a down there so we are going to remain with negative B squared plus form SC over for n so what it means there is that whenever I'm trying to find this part we are trying to come up with this a X Plus p squared plus Q so whenever I'm trying to come up with P meaning that this p is the same as B over 2m I get this okay I get this so now it is the same thing now whenever so this is going to be our Turning Point our turning point now is going to be this so our Turning Point here is going to be our turning point is going to be B over 2A comma negative B squared okay then we're going to have plus 4 SC over 40. so this is going to be our Turning Point then now whenever we're trying to find the line of symmetry we just equate this part which is inside equal to 0 and find the value of x that is the line of symmetry so let's see let's say we have a question we have this they are saying using Computing Square methods to find the turning point x and y-intercept and sketch the graph and also find the line of symmetry so we have the first question here which is saying um so we have this one so the first point we said is to factor out the m okay so if I factor out a here which is going to be three I'm going to have x squared plus here I'm going to have 2X minus here I'm going to have um 4 over 3. now from here what else do we have we say do we need first now to find the half of this so this part here so I'm going to find the half of that one so we're going to remain with this we're going to have 2X half of this is one so we say plus is going to be 1 now we Square it the same thing is going to be minus one now we Square it again minus 4 over 3. and then I do this now from here I know that I'm supposed to ignore this I'm going to get this part and this pattern so I'm going to have three X Plus 1 I squared minus 1 squared is going to be 1 minus 4 over 3. this is what we have now if we have negative 1 minus four over three it's the same as we have negative 3 over 3 minus 4 over 3. because negative 3 over 3 is negative one so this is going to give us negative 7 over 3. so this part is going to be three then X Plus 1 squared minus 7 over 3. okay so I have this now I can distribute now three okay I'll get rid of this I can now distribute three so if I distribute 3 I'm going to have three X plus one squared minus is going to be three seven over three now three and three can go okay if we can go we are going to have this so mean that what we have here is a in the form of a X Plus p plus Q so we can see that our Turning Point here is going to be our Turning Point here is going to be what we get here is um we get the opposite of this okay so in short what I'm trying to say is we equate to this equal to zero solve for x x is going to be negative one so that is the value of x so we have negative one comma negative seven these are the turning points okay now from there what else do we do we know we can see that negative 1 comma negative seven that is our Turning Point that's where the graph is going to to 10. now from here we can also find the what the X and y-intercept now to find the line of symmetry this is the line of symmetry you just equate what is inside equal to zero and find the value of x that is the line of symmetry okay now to find the x-intercept or let's start with the y-intercept to find the y-intercept uh if you want to find the y-intercept intercept set x equal to zero meaning that in this equation where there's X let's put 0. so our Y is going to be negative 4. okay so we're going to have 0 comma negative four that would be our Y intercept to find the x intercept same way if we equate um y to be equal to zero so let's just put this here we're saying that our Turning Point our turning point is negative one comma negative one seven what else do we know we also know the y-intercept okay the y-intercept the y-intercept were found that is zero comma in X 0 then negative form to find the x-intercept what we have to do is set y equal to zero then we are going to find the x-intercept meaning that what we have there is we have 3 x squared plus C six x minus 4 is going to be equal to zero so for x this is not factorizable what we're going to do we are going to use the bus method so we're going to use x is equal to negative B this B squared minus 4M see everything divided by 2m so we have X is equal to our B is a 6 so we're going to have negative 6 plus or minus we're going to have B squared is going to be 6 squared minus 4 a is 3. um C is negative four so we have this 2 times 3. so X is going to be equal to negative 6 plus or minus inside here we are going to have 36 then this is going to be plus so we have 12 and then we have 12 times 4 . so 12 times 4 which is 48 so we have plus 48 here then you have everything divided by what six okay so we can get rid of this the top part here okay then now what else we can see that 48 plus um 36 we are getting 84. so we're going to have X is going to be equal to this we're going to have 84 over 6. now we don't have the square root the square root of 24. we don't have the square root of 24 but if you use a calculator you can find the square root of 24 but now the issue is if you're not allowed to use calculators you can leave your answer inside so here I'm going to use a calculator okay so we're going to say that what is the square root of 84. square root of 84 is 9.17 okay so we're going to have X but if you're not allowed to use a calculator you can leave your answer in science so it's going to be we are going to have 9 .17 everything divided by six so we're going to have two values of x x is going to be equal to negative 6 Plus 9.17 over 6 or X will be equal to negative 6 minus 9.17 over 6. so what will be the values of X now when Y is equal to zero so we're going to have X will be equal to we have negative 6 Plus in 9.17 okay so I divide this by 6 I'm getting 0.1 I'm getting a 0.53 0.53 I've just launched it off then another one is X is equal to negative 6 minus nine pointy one seven I divide it by 6. I'm getting a negative two point five three so now we have the x intercept when the Y is zero we have 0.5 3 comma 0 and also negative 2 comma negative 2.53 comma zero so these are the x-intercept and those are what they were intercept now how do we find how do we sketch now the graph so it is very very very simple now since we have everything it's just the amount of us plugging in the values here so we have this and this now the value of a remember that when the value of a is less than zero our graph is going to be at its maximum okay it's going to to Cave like this if the value of a is greater than 0 like in this case our value of a is greater than zero the graph is going to cause its minimum point we're going to have this that is very very important so before you even graph uh you even sketch the graph you'll be able to know that the graph is going to be at its maximum or minimum so if the a is less than zero just know that the graph is going to be like this okay when the a is greater than 0 the graph will have its minimum point that is also very very important now we have the values we can see that we have let's say we have a one here a one a two and a three so I have one two three so here is also have um a negative one negative one negative two negative three okay so we have negative one negative two negative three so we have a one two three one two three here we're going to have negative one negative two negative three negative four because the turning point is that negative seven I need to go all the way to negative seven negative five negative six negative seven here now what we have to know is um that is our turning points negative one comma negative seven is this point here okay then we have the X the y-intercept where the graph is going to cut is negative four so this is I want negative four here okay when the graph rest there is going to turn at negative form in the y axis now in the x axis we have zero comma negative zero comma uh five point three which is somewhere here okay then you have also negative 2.53 which is somewhere here okay comma zero so meaning that the graph is supposed to come from here it goes all the way to this point then it rests here then it goes where there's four and it goes there so that is our graph of f of x is equal to 3x squared plus 6 x minus four now the line of symmetry we said it is at 1 Negative X is equal to negative one so this is our line of symmetry just put this so that is our line of symmetry at X is equal to negative y negative one that is also very very important so this is how you can find the x-intercept the y-intercept the turning point and also the line of symmetry and also how to sketch the graph using the Computing Square method yeah that is very very important so we also have a question question two here which we can just solve you can pause the video and work it out and see the answer which you're going to come up with okay so you have the second question which is saying f of x is equal to x squared minus 4X remember we said is to factor out X sorry to factor out the value of M so in this in this case a is one so if you want you can just put one then you have x squared minus four uh X we don't have anything here it's zero okay so we need to find the half of this that is also very very important so we have one so we have x squared minus form X now it's going to be plus the half of this what is the half of negative four it is negative two so I'm going to put a negative what I'm going to put a negative 2 there okay so I'm going to put a negative 2. here I'm going to put a negative I Square minus I need to get the same thing I also Square it then I have this okay so what we're going to have here now we're going to have one we're going to have we ignore this that's what we say to ignore this so we just get uh this and this okay so we're going to have X minus 2 square to minus negative 2 square this 4 so we're going to have this meaning that we have now 1 x minus 2 then 4. now to find the turning point we say that the turning point I can get rid of this I need I no longer need that the turning point this value is supposed to be the opposite okay so X minus 2 is equal to zero find the value of x that would be our first standing point the turning point in x axis so it's going to be 2 comma at negative four this one you just get the way it is if it is negative just get Negative if it is positive just get it positive but this part here if this is negative get a positive if it is positive get a negative in short just get what is inside set them equal to zero then so for x at the same time that is the line of symmetry okay then now what we need to find is that we need to find the x and y intercept how do we find the X and y-intercept okay if it is factorizable if I want to find the y-intercept what I need to do is um to find the y-intercept x equal to zero okay that is in y-intercept X has to be equal to zero meaning that in F of 0 is going to be 0 squared minus 4 then 0. so this is going to give us 0 comma zero okay so that is our y-intercept our y-intercept is zero comma zero now to find the x intercept set y equal to zero okay that will y has to be equal to zero this is our y our x-intercept so it's going to be x squared minus for X is equal to zero I can factor out x x minus four is equal to zero so X would be equal to zero at the same time x minus 4 will be equal to zero X is going to be equal to 4 there is going to be four so we have two value two uh x intercept so we have x intercept which is going to be 0 comma zero we are going to have also four comma 0. after finding this the line of symmetry we know already that is going to be 2 because we have to set that one equal to zero so the line of symmetry line of symmetry is going to be at X is equal to 2. so to sketch the graph now we are going to see that this is going to be like this okay let's put it in this side okay at the same time we know that our air was greater than zero it was one actually so the graph is supposed to be having its own minimum that's why it is negative this part here okay so before we even graph you even sketch the graph you you know that the graph is supposed to to be like this or supposed to be like this that is also very very important so now what we're going to have to do here is that we're going to have our values here we're going to have one two three four five so one two three four five then I'm going to have my values here one two three four five okay so negative 1 negative 2 negative 3 negative 4 negative five so here I'm going to have one two three four this is the y axis this is the x axis okay so I'm going to have one two three four I'm going to have negative one negative two negative three negative four that is just Okay negative one negative two negative three negative four now from here I need to find the the Y the y-intercept is zero comma zero zero comma zero is this point then x intercept again we have zero comma zero so it's going to be at the same point okay then the turning point is two or the way to four so it's this point here that's the turning point that's the Turning Point that's where the graph is going to tell now we have also the line of cement is at the same point then you have uh four comma zero is this point so the graph is supposed to come here and it comes here it rests then it goes there so this is my graph of f of x is equal to x squared minus 4X the line of symmetry is at this point is at uh two comma negative four so this is a t two comma negative four so you can say that this point is at the negative or is at 2 comma four two comma negative 4. so this is how you go up with um the Computing of square method we are going to talk about Computing Square methods which involves modulus so how do we sketch now we've got a question which is saying complete the square of each of the following quadratic functions hence sketch each graph indicating the turning point and the intercept and write down the equation of each line of symmetry so as you can see part one has got modulus but there is no negative but two there's modulus but there's negative I think if you manage to check the previous video where we introduced the Computing Square method we talked about how to find the line of symmetry how to find the X and y-intercept and also how to find what the turning point now here one just to talk about what if you have been given the modulus how do you sketch let's talk about uh how to use remainder theorem to find the remainder so I have two questions with me the first question is saying um 2X to the power three minus 3x squared plus 4X Plus 5. then they're saying if it is divisible by this