1.1 review of functions the basic idea of a function is that you pick a number you apply an operation to it such as plus minus divide multiply or square root Etc and then it becomes a new number so if I pick the number two and apply two operations to it such as multiply by two add one so 2 * 2 is four add one is five so the number two when the function is applied to it becomes the number five and becomes therefore a new number so what you're doing with functions is you're sending one number to another number with a rule in between those two numbers that gets it sent to the new number now it seems inherent since you get to pick whatever number you want to start with that that number is the independent number and that is what we call X the result the five is the dependent number and so what is it dependent on well it's dependent on not only what you chose to plug in the number two but it's also dependent on the rule which was add to sorry multiply by two and add one that you apply to it okay now if you take that same rule which again was multiply by two add one and you apply it to any number that you pick meaning it doesn't have to be the number two it could be the number three or the number four or the number five then if you were to graph those inputs and outputs on an XY coordinate plane what will happen is it'll result in a pattern and that pattern you get depends on the operations you choose for the rule so multiplying by two and adding one will create one type of pattern throughout the section you'll you'll see various rules that creates different types of pattern so by definition a function f consists of a set of inputs and a set of outputs your x's and your y's your domain and your range and a rule for assigning each input to exactly one output so just a second ago again I said my rule was times 2 and my second rule was add one so you take each input number times it by two and add one and you get an output number and so the set of inputs is called the domain as I mentioned just a second ago which is all the numbers you get to choose to plug in the set of outputs is called the range all the answers you get after you apply the rule and so let's specifically look at this times two and add one and see what kind of pattern that creat creates and so you start with the number X cuz you get to pick that number and the rule says that you multiply by two and then you add one and then the result of that is my output which we will call Y or you could call that f ofx as well those two are the same thing Y and F ofx just a different way of writing it and so if I pick a bunch of inputs and then get the outputs after apply after I apply the two rules let's see what pattern that creates so let's create a x y chart and let's plug some numbers in such as -2 -1 0 1 2 so again the rule is times 2 add 1 I'm going to apply that to each of those so let's start with -2 * 2 -4 add 1 -3 -1 * 2 -2 add 1 1 0 * 2 is 0 add 1 is 1 1 * 2 is 2 add 1 is 3 and then finally 2 * 2 is 4 add one is five now let's see what kind of pattern that creates 0 1 1 3 2 five so right two up five 1 1 left 1 down one and then left two -2 down three you'll notice every single one of those line up because the pattern it creates is a line in fact any rules that are just your basic operations times multiply add divide as long as they're only used once it creates a straight line and so a simple way to think about all that is if the highest power here is a one by default it's called a linear function and the pattern is is a line now let's look at all the different classes of functions that you should have memorized to do well in this course the first one we just did it is a linear function in other words the highest power is a one I'm going to quickly do some inputs and outputs there this function f ofx = x this is called a parent function because it's the simplest version of a linear function in this case it's a it's the simplest version of any function because whatever you plug into it is your actual output if you plug in a one you get a one if you plug in a two you get a two it's called the do nothing function because when you plug in a -2 you do nothing and you get -2 when you plug in negative 1 you do nothing and you get negative 1 when you plug in zero doing nothing into it gets you zero plug in one you get one plug in two you get two you'll notice all those will line up once again in a straight line I even added a point for you there 3 and3 and 3 3 because we didn't do anything to it anytime you think about graphing what you're really doing is you're manipulating this basic straight line you're bending it you're stretching it you're pushing it up in other words if you look at X squar that's obviously not as basic as X however what you're really doing is you're taking this line that we created which is the do nothing function and you're bending it to create the new function and so let's do some inputs and outputs here we get -2 if you square that you get four because the rule is timesing by itself 1 you get 1 0 0 1 2 is 1 2^ 2ar is 4 and if I graph that I'll get 0 0 1 1 2 4 - 1 1 -24 and so this is the most basic version of a parabola and you you should have this shape memorized now one of the things I wanted to say up here because this is important as well is we're going to talk a lot in this class about domain and range visually if you have a graph domain and range is very easy domain is left to right in other words how far does the graph go left and right and so the answer here is negative Infinity to Infinity how far does the ground the graph range go down to up this graph goes down to negative Infinity and up to positive Infinity now if I look at domain and range here domain left to right it goes negative Infinity to positive Infinity which means you can plug anything into the function to get an answer range though is a bit limited here you'll notice that it does not go down forever in fact the lowest it goes is zero and so because zero is included in that we'll put a bracket and it's 0 to Infinity because it goes up forever all right let's look at Square < TK of X which is the inverse or the reverse of X2 and so we'll go go ahead and pick some values again this is one though where when you pick the values you kind of got to think about it ahead of time here because you cannot plug anything you want into a square root in fact you can't take the square root of 1 or -2 that doesn't get you a real number and so the smallest number that gets you a real number here that we can pick is zero and the square Ro of0 is Z one is a nice number to pick because the square root of one is 1 two you don't want to pick two there's an answer for sure but it's a decimal so what we're going to do is instead pick the next perfect square of four because the square root of four is two and then we can pick nine because the square Ro of 9 is three let's graph those 0 0 1 1 42 and 93 and so the general shape of this function function looks like that again let's do domain and range which is domain is limited here the smallest number you can pick is zero and then up to Infinity you can pick anything range also limited here the lowest point on the graph is zero going this direction and the highest point is well it does keep going up forever so what you get is you get also 0 to Infinity again important to remember the shape and then visually you quickly know the domain and range let's look at another one of the important functions to remember which is X cubed if I pick some values for X cubed here I will pick -2 -1 0 1 and two if I Cube zero I get zero if I Cube one I get one 2 Cub 2 * 2 * 2 is 8 here I get -1 cubed is - 1 and then 8 and so the shape of this graph looks something like this so -28 so you get that and then it flattens out in the center and then goes up like that so that's the general shape there domain left to right it goes left and right forever negative Infinity to positive Infinity range it goes down and up forever because you have negative Infinity to positive Infinity as well okay so we're almost through the basic functions here that you want to have memorized so the next one the exponential function the exponential function is repeated multiplication so in this case 2 the X which is 2 * 2 * 2 whatever you plug in for X so let's pick some values here -2- 1 0 1 2 to get the basic shape let's do the easy ones first to the 0o anything to the zero power is one 2 to the 1 power is 2 2^ 2 is 4 if I do two to the negative one a reminder here is the way you take care of the negative is you put the two on the other side of the fraction bar and you can do that if you take this power and make it positive and then since there's nothing left on top we'll put a one so the answer here is2 and then finally 2 to the not4 but rather -2 power we use the same trick but the two on the bottom that creates the power to be positive so 1 over two squ is one fourth and so we get 0 1 1 two 2 4 we get neg 1 12 and we get 1/4 -2 and so this graph looks like [Music] that like that and there's an asint toot on the x-axis because it'll get infinitely close to that but never quite touch it and so the domain main here is well left to right so the graph does go left and right forever so negative Infinity to positive Infinity the range how far does it go down well it never crosses the ASM toote or even touches it and so we'll say parency not including zero to Infinity the next function is the reverse function or the inverse function of the exponential function to the x is log base 2 of X for now we're going to use the trick that inverse means reverse and so what do I mean by reverse well when you do an inverse all you have to do is swap the X and the Y values and so that's what I'm going to do this one is 1/4 comma -2 uh 12 comma - 1 1 0 2 1 and 42 let's do a quick graph of that 4 2 2 1 1 Z and then 12 comma -1 and then 1/4 comma -2 so we get a shape that looks like this with an Asm toote on the y axis and so domain range the domain is going to be left to right well it doesn't go left past zero so we'll say 0 to infinity and the range is going to be negative Infinity to positive Infinity again if you go through the book it doesn't really do this part it doesn't show you all these basic graphs but I think they're very important so I included them so let me scroll down here and we'll do one more which is 1 /x the very last one here that really you should just have memorized it makes your life a lot easier let's pick some values that I know to to work well here and so you wouldn't necessarily know these values ahead of time but I know good ones to pick here so let's start with um - one2 and then - 1 and then 0 1 and then positive 1/2 or some good values to pick that there because if I do well let's do the easy ones first 1 over 1 is 1 1 over Z oh okay I can't pick zero and so that's one of the things you have to remember in the domain that you cannot divide by zero so let's do the other values first this is negative 1 if I do 1 over 12 remember invert and multiply so 1 over 1 times flip that around 2 over one so I get an answer of two so this one will be -2 and so let's pick another value here what would be another maybe reasonable one to pick well let's graph it first and go from there so if I do 1 one and I do 12 comma let's see 1 12 comma 2 which is right there and so maybe if I pick a value let's add one more here maybe if I pick a value of two your answer is 1/2 and so two and 1/2 that gives me a little bit idea of what this looks like that's right there and so it's going to curve something like this go out that way and go up that way and then on the other side again11 - 1/2 comma -2 and let's again let's pick -2 as an extra value here if I pick -2 as a value I get -12 so -22 so this one is going to look something like this on this side and then shoot down towards the Y AIS and so there's actually two ASM tootes here one is the one on the x axis and the other one's the one on the Y AIS and so last thing here domain is going to be well it goes left forever so that's negative Infinity but then if I kind of go to the right there's an Asm toote on the Y AIS so you cannot include zero notice earlier I couldn't plug zero in So Close parthy Union Union just means it's going to start back up again back at zero with a parenthese not including zero and it's going to go right forever to infinity and actually the range here does the same thing it goes down forever all the way up until the x- axis which is zero Union 02 Infinity again even if you don't know them yet that's something you want to look at constantly all semester long the more you know these graphs the easier life will be when you're doing the calculus all right so let's jump into example one here and we can use some of our knowledge about these graphs to do it in this case you won't necessarily need that knowledge but you will in further problems in the sections example one evaluating a function simply just means plugging in the number they're asking you to plug in for X all right so F of -2 so that means F of -2 equals well that just simply means wherever you see an X plug it in -2 before you do that my advice is write it like this write 3 parentheses squar in other words instead of X write a parentheses it makes for much less mistakes to include the parthy plus 2 parthy minus one and then you can come back and plug in the -2 so now we can simplify that -22 is 4 so that's 3 * 4 - 2 * -2 which is -4 -1 all right so 3 * 4 this is 12 12 - 4 is 8 - 1 is 7 all right let's do it with another one let's do F of -2 which is 3 * something 2quared + 2 * something minus one then come back plug in your < tk2 and your < tk2 remember the square root and the squared cancel each other out because they're inverses they reverse each other so what you have left here is 3 * the 2 because the operations reverse each other plus 2 2 < tk2 - 1 so 6 + 2 < tk2 - 1 which equal 5 + 2 < tk2 so now let's look at part C which is f of a + H so F of a plus h equals and so again we're going to plug in a plus h wherever we see an X so we get 3 * a + h^ 2ar + 2 * a + H minus 1 so we get 3 * a + h * a + H + 2 a + 2 H if I distribute here - 1 so now we're going to do foil on this part which is first which is a * a so a 2 the outside a * H the inside again H * a but we can write it in the same order a * h plus the last h^ 2 + 2 a + 2 H -1 okay distribute three we can also combine like terms here that right there is 2 ah so if I distribute now I get 3 a 2ar then 3 * the 2 a is + 6 a h + 3 h^ 2 + 2 a + 2 H -1 finally we can can combine like terms if there is any let's see if there is any here um we have 3 a^2 6 ah+ 3 h^ 2 then + 2 a + 2 H -1 it doesn't seem like I can combine anything so that would be our final answer example two find the domain and range for each of the following functions determine the domain and range okay so a f ofx x = X - 4^2 + 5 to figure out the domain I can actually look at it and determine what restrictions have to be put on the operations in the function for range it's best that we go ahead and graph the function using the toolkit functions and then the Transformations so we can quickly see the lowest and highest point once we graph it we'll be able to verify our domain as well so when I look at number a I look at specifically the operations that are going on which which is minus you can subtract any two numbers and get an answer therefore subtraction does not have any restrictions on the domain addition again you can add any two numbers and get an answer so addition has no restriction so we don't have to worry about the addition part squared just simply means times itself if you take a number and multiply it times itself again there's no restrictions on that there's no two numbers you can multiply and somehow not get an answer so we can say here that the domain is infinity to Infinity okay let's look at the range of a well to look at the range what we need to quickly do is draw a little graph using our toolkit function we know it's squared so what it starts off at is the parent function for that looks something like this just x s and so if I take those dots and move them which way well this we do the opposite of that whatever makes it zero so we go right four and then here we go up five and so if I take those dots and go right 1 2 three four and up five I can still easily see the graph enough to determine what the range is the range is the lowest point to the highest point so what's the lowest output on the graph it's five that's the lowest answer I can get what's the highest point it goes up forever so the range is not parenthese but rather bracket since five is included in the range to Infinity let's look at B > 3x + 2 - 1 okay so first let's determine domain by doing the same thing I did early let's look at the operations the operations are times three you can multiply any two numbers together and get an answer plus two you can add any numbers and get an answer minus one no restriction there because multiplication a division sorry multiplication addition and subtraction have no restrictions on what numbers you can put into them so what's the last operation the last operation is the actual square root it turns out that the square root does have a limitation at what you can plug into it what are you taking the square root of well we are actually taking the square root of the entire piece underneath now the only restriction on the square is that whatever you plug into it it cannot be negative because you cannot take the square root of a negative number so 3x + 2 therefore cannot be negative again 3x + 2 cannot be negative so if it can't be negative what can it be well 3x + 2 can be bigger than or equal to zero because if whatever answer you get for 3x + 2 if the answer is bigger than or equal to zero then you can take the square root of it so let's actually move through the math to figure out what x value makes this entire piece bigger than or equal to zero so you can then take the square root so if I minus two from both sides and I divide by three on both sides I get -2/3 that means this is the barrier that's the smallest number I can plug in and still get an answer in fact when I plug in -2/3 into there my answer will be zero if I pick anything small smaller it won't work if I pick anything bigger all of it will work because it'll produce a positive answer okay that means the domain is including -2/3 because that's the one that made it zero anything you pick that's bigger than -2/3 will make it positive so you can take the square root so that's the domain all right visually if I go to the graph how do I graph that well I start with the square root function which is square of 1's one square of four is two so those are some dots I can use how does it move around the moving around comes from that 3x + 2 how do you know how it moves around well okay the way that moves around is you have to say 3x + 2 is = to 0 3x is = to -2 and X = -23 this is the way it moves we actually had done that over there already that's the way it moves whatever makes it zero is the way it move so -2/3 so what I do is I take my dots and I move it left 2/3 right there and the second piece of that is I subtract one so I take that and I subtract one so I'm right there now I can draw my function so once I've drawn my function then where I have the -1 is the lowest point and then what's the highest point well this will keep going up and up forever so the range is going to be NE 1 to Infinity all right finally the last one approach it the same way which is what operations are there minus no limitations on that we're still at negative Infinity to positive Infinity but there is a division operation there what are you dividing by you're dividing by xus 2 what can you not divide by division actually has a limitation you cannot divide by zero that doesn't produce an answer if you divide by zero the answer is undefined and that means there's no answer okay that means xus 2 if I can't divide by zero x - 2 cannot equal Z so X cannot equal 2 okay that means the domain is negative Infinity up to two but not including two Union starting at two but not including two to Infinity then finally we graph that by using the toolkit function toolkit function for one overx says this here's some values that I can plug in one and then if I plug in 1/2 I get two if I plug in two I get 1/2 so it looks something like that on the other side I get here again I'm just plugging in values here and here so those are the two curves so what am I going to do with that well you have to multiply it by three that's one thing you have to do because there's a three on top but we're not worried about that part in the domain AR range because that actually doesn't make it move left or right um the xus 2 since that's with the X that's going to what makes that zero positive two so I have to take this and move it right one two like that and then there was an ASM toote at the x-axis now the ASM toote is there and this graph looks like this and like this all right so now we can say that the range the lowest point is negative Infinity all the way to where is there no answer well there's no answer at zero because it stops there for a second Union 0 to Infinity because it keeps going up forever the vertical line test determines whether something is or isn't a function so what makes something a function well what make something a function is that for each input you can only have exactly one output okay and so the vertical line test allows us to do that so for example if you had a point here which in itself is a function even though there's only an output for two so input two output one that can be the entire function that's a very basic function but it is a function because for the one input there's only one output now if I simply just add a second point at this point it's a relation it's no longer a function and the reason is because the input of two has an output of one but also has an output of two in other words it doesn't pass the vertical line test because a vertical line that I'm going to draw touches the function twice and when it touches it twice it's not a function it says it says up there all equations can be graphed only the graphs where any vertical line placed anywhere on the graph intersects the graph exactly once as a function so that is not a function because that vertical line touches the graph twice um some other things that are not functions you could easily have a circle that is an equation or a relation but it is not a function because again a vertical line pretty much anywhere you draw it almost would hit the circle twice okay so things that would be functions well we have a bunch of them up here all these that I did that's why it was a review of functions are functions because any vertical line would hit any of these graphs only exactly once and so if I come back down here um I could draw this curve I could add a point here I could draw a line from here to here this is kind of like a piecewise function the whole thing though is still a function because if I take any vertical line and drag it along at any point on this graph it only touches the graph once right it's different here because it touches it twice so that is the vertical line test all right x and y intercepts all right to keep this simple how do you find x and y intercepts well if you have a graph so let's say we have a line and you kind of I'll drag it over here where is the x intercept well the answer is the x intercept that crosses the x axis at the point 1 comma 0 okay now if I move it now it crosses at -2 comma 0 but notice a certain consistency here it seems like no matter where I put that line the y coordinate or the F ofx coordinate is always equal to zero so how do you find an X intercept of a function you let the y-coordinate the F ofx equal Z and then you just simply solve it for X all right let's do the same thing but this time let's look at the Y intercept if I draw a line and the Y intercept is clearly at the point 0a 2 if I move it down to here now the Y intercept is 0 comma -3 this time the consistency is that the x coordinate is 0o so how do you find a y intercept you let X be equal to zero and you solve for y or for f of x let's keep going consider this function find all the of f all right what does the zeros of f actually mean well the zeros is actually the same thing visually it's the same thing as the X intercepts in other words where is the function equal to zero that's what it means to be a zero of f where is f ofx equal to zero that's what the Zer and again that coordinates with the X intercepts because f ofx equals z so what am I going to do I'm going to say okay how do I find all the zeros by saying 0 = 4x + 2 and then I solve it for X so min-2 on both sides -2 = 4X and divide by 4 so you get -24 equal x reduce it I get - one2 and so what happened here is I put a Zer in for f ofx and my answer was -2 0 for y if I graph that that means it would cross the xaxis right there at - 12 comma 0 find any y intercepts all right so for a y intercept we discovered up here that you need to let x equal Z so let's do that so F of to find a y intercept let x equals 0 that's where I'm putting it right there so -4 * 0 + 2 equals 2 and so what did I do I plug the zero in for x X and my answer was 2 for y you probably knew that already because this is mx + b form and this is always the Y intercept so 0 comma 2 and it says sketch a graph well I know it's a line now and for a line I only need two points and so those two points will suffice so if I graph that those are my two points okay let's keep going consider the function s < TK x + 3 + on same directions find all the zeros of f the Zer again Zer are where is the function equal to Z where is f ofx equal to Z those are the X intercepts so let's do that 0 equals < tkx + 3 + 1 how do you solve functions you get the parent function by itself first you get the in this case the square root root part I guess I shouldn't specifically say the parent function but I should say the rule by itself the rule that makes the particular shape in other words there's a lot of rules going on here there's like X+ 3 plus three is a rule plus one is a rule but the parent type of function here is the square root and so with any function you want to get that piece by itself first the more complicated piece and so getting the square root by itself first means subtracting one from both sides and so you get 1 = < TK of x + 3 okay so once you get the rule the more complicated part of the rule the square root by itself then you reverse it by doing the inverse of the function so the reverse of a square root is to square both sides now you have to be a little bit careful here because sometimes you have to stop in math and think okay is this even possible can I take the square root of a number no matter what it is and get an answer of negative 1 and the answer is you cannot take the square root of a number and get a negative 1 at least not a real number and so what you should do is not keep going not Square both sides and just go there is actually no solution here that means visually this graph never crosses the X AIS okay so how about the Y intercepts well let's see if we can do that F of 0 equals > 0 + 3 + 1 and so the answer here isqu of 3 + 1 that is a decimal answer so something like 1.7 + 1 or approximately 2.7 so when I plugged in the zero the approximate answer was 2.7 and so that is somewhere over here okay so that's the Y intercept um you could write it as a decimal or you could leave it < tk3 + 1 sketch it okay well we know what the square root graph looks like so let's use that to help us out and let's do some Transformations so the square root graph looks like this those are three dots on the square root graph square of one square of0 square of 4 and this says left three up one so let's do that left one two 3 up one notice that black dot Falls right in line with the other ones as it should and so now I'm going to graph that like that okay and that is it let's keep going find all the zeros of this function again what is zer mean zeros mean set the function equal to Z set F ofx equal to0 so 0 equals x Cub - 5x^2 + 6x this requires some algebra skills here of factoring that's generally how you solve equations you set them equal to zero and then you try to factor if it doesn't Factor then it becomes a little more complicated let's see if we can Factor this the first rule of factoring is pull out a common factor so 0 equals what's common to each of these is an X so what's left is x^2 - 5x + 6 so 0 = x and now since we have a trinomial you should hopefully remember the idea of factoring this which is this is your first outside plus inside and last so we're going to factor so for the first I need x and x x * X is X2 for 6 and 2 I'm going to guess at that and I'm going to guess 3 * 2 and now I've got to check that the outside plus the inside the outside which is 2x plus the inside which is 3x gives me -5x I can do that if each of these are negative so I'll make that negative and that negative and so now what I have is x * x - 3 * x - 2 is equal to Z so how would I make that be equal to zero well either this one could be zero if that one's zero then everything will be zero if this one's zero what would what would what is needed to require that to be zero the answer is 3 3 - 3 is zero or if this one's zero x = 2 so visually the graph crosses the x-axis there at 3 and at two and so you get 0 0 3 0 and two Z as my all my zeros are my X intercepts now I don't have to graph it you can check that on Desmos or some graphing calculator to graph it on I just kind of visually wanted to show you that is where the graph crosses the x-axis probably this thing will look something like this again approximately not not exactly it'll probably look something like that bounce back down and bounce back up you can check me that is again that's not exact it's just using the fact that I know the X intercepts find the height of a Free Falling object if a ball is dropped from height of 100 feet it's height s meaning how high it is above the ground at time T how many seconds have gone by since you dropped the ball is this function so two things are going on there t is an input how much time has gone by S of T is an output how high is the ball above the ground okay so the domain is restricted to the interval Zer to C that just means time can't be negative it has to be some positive time until it hits the ground which is at the point C where tal 0 is the time when the ball is dropped and tal C is the time when the ball hits the ground okay so create a table showing the height s of T when T is 0.51 those are all inputs so in other words come over here and plug in zero plug in 0.5 plug in one plug in 1.5 2 and 2.5 in into which function this function right here and get an answer for that and so what you're doing is plugging each one in the easiest one is obviously zero because you can get -16 * 0^ 2 + 100 and that just leaves you with 100 I'm quickly going to go through this and then I'll come back and plug in those values the next one would be -16 * 0.5 SAR + 100 you can use your calculator to do that I will show you one of them and how to type that in here so -16 * 0.52 so6 and so you're going to go ahead and an quick an easy way to do that and I'll show you that here on the calculator is to actually type in um I know it's t but you have to use x here on the calculator so I'm going to put x^ sared and then plus 100 now right now there's some other value in there for X but I have to hit enter because it'll have that equation in in there and so if I plug in zero I'll give you an example how to do that I already know what the answer is but look zero and then you hit the store button which is two up from the bottom left here store and then X it'll store zero into X and then to pull up my equation again I can hit second enter second enter again it'll pull it up and now it'll plug in that zero so I'm going to do that for all the five numbers so if I go 0.5 store s X enter now that's stored in for X so second enter second enter pull up the equation and now it plugs in 05 so the answer is 96 so I'm going to do that for each one and then I'm going to come back okay we're back and I plugged all those in and figured out all those answers with my calculator now let's put that on the graph so 0 100 0.596 which would be probably about right there 184 again I'm just being approximate here 84 1.5 and then 64 would probably be somewhere around there and then 236 would be around there and then 2.5 comma 0 would be on the ground so the graph here that models a ball falling oh let's try that one more time now again this isn't actually the trajectory of the ball although it could be if you kind of threw it out you could just drop the ball straight down this tells you just inputs and outputs meaning your input is and I visually I've got this wrong here a little bit this should be T I'll go ahead and correct that that should be T and then your output so this is time and then your output is the height above the ground now from the graph we can figure a lot out here and so it says says we created a table okay we did that then find the time C when the ball hits the ground well from the data in the graph obviously at 2.5 seconds C = 2.5 seconds the ball hits the ground and then it says sketch the graph of s well we just did sketch the graph of s so this is one of those problems where we really use the table and the graph to figure everything out okay um we could have used the equation as well but we didn't in this one we used the graph and the table let's keep moving combining functions using mathematical operations what does f plus G of x mean it means you're adding two functions together it really just means f ofx plus G ofx this one really just means F ofx minus G ofx this is f ofx time G ofx multiplying the two functions and then the last one is f ofx ided by G ofx dividing the two functions okay so if I add these two functions together I get 2x - 3 + x^2 - 1 combine like terms so I get x^2 I'll put the biggest power first + 2x and then - 3 - 1 is -4 If I subtract I have to be a little more careful I have to go 2x - 3 and then minus parthy x^2 - 1 because this minus will distribute that's like a minus one to both of these so what I get is 2x - 3 that part doesn't change but - x^2 + 1 so what I get here is combining like terms is x^2 + 2x and then I can combine -3 and + 1 to -2 all right multiplying these together 2x - 3 * x^2 -1 four I do that 2x * x^2 is 2x cubed outside - 2x inside - 3x^2 and finally last is + 3 so that's what I get when I multiply them together and then finally F ofx which is again 2x - 3 for the division divided x^2-1 you can't really do much here this is I mean I could do this I could Factor the bottom into x -1 x + 1 but again that's it there's nothing I can cancel there so this is my final answer so some basic mathematical operations with functions now let's move to function composition here is the basic premise of comp composite functions pick the number one plug it into the function f ofx = x + 2 so the answer is 2 + 1 is 3 now you plug the answer three into a second function let's say G ofx = x^2 and the answer is 3^2 = 9 so in other words one went to three and then three went to nine so one went to three because the the function was plus two and then I plugged three into a different function and got nine You're simply starting with a number applying a rule getting a new number then applying a second rule to the new number to achieve another number let's do the above example with variables pick a number X plug it into f ofx = x + 2 well if I plug X into there which that's X you get x + 2 now plug x + 2 into G ofx = x^2 where am I going to plug it into well you always plug stuff in for x and so what you get is you plug it in there and there you get x + 2^ 2 okay let's do an example consider the functions X2 + 1 and 1/x let's go ahead and do G of f of X okay what does this actually mean the way you write it is like this G of f ofx in other words you're plugging F ofx into the G function all right so let's go ahead and do that and the way to do that is you go okay I highlighted the F ofx function so that's what I'm going to highlight and that is going to be plugged into the G function for x and so what I get is I get G of f ofx equals again I'm taking this and plugging it in for X into the other function equals 1 over x^2 + 1 and that's it okay evaluate G of f of four well I already have G of f all I'm simply now doing is plugging a four into there which means I plug a four into there and so what I get is 1 over 4^ 2 + 1 which is 1 over 16 + 1 which is 1 over 17 okay evaluate G of f of - one2 again it just simply means I already have G of f plug it in so 1 over2 2 + 1 which is 1 over -2 * -2 is4 + 1 so I get 1 over 1 and 1/4 1 and 1/4 is 54s so I can flip that around and multiply and I get four fifths okay now when you do G of f you will not get the same answer as when you do F of G because this time you're plugging the G function that's the inside function into the F function let's see if we can do that so I'm going to erase these highlights here I'm going to take take this function and plug it into there so 1x^2 so I get F of g ofx equals the outside function is that one so it's something squared + one what am I plugging in 1/x so the answer here is 1 over cuz 1^ 2 is 1 x^2 + 1 and that's it now how do I plug numbers into that well if I plug a four in that just means plug in a four so F of G of four I already have the function so I get 1 over 4^ 2 + 1 which is 1 over 16 + one which is 1 and one 16 so you get the idea here I could also plug in the negative - one2 same idea let's move on here and look at absolute value functions one that comes up quite a bit as well in calculus what is the difference between yal X and Y absolute value of x for all positive inputs the answer is the same okay nothing changes you plug in a two you get a two three you get a three absolute value that's the same thing that this means part of the graphs must be exactly the same but for negative inputs yal X is the opposite output of yal absolute value of x because on this one when you plug in a negative you get a negative but on this one when you plug in a negative you get a positive okay so if you plug in -2 for the answer to Y X is -2 but if you plug a -2 and the y equal absolute value of x the answer is positive2 okay in other words for negative inputs yal absolute value of x is the same as the function y = negx instead of positive X because it turns them all negative okay since the double negative makes the values positive that is is -2 = 2 that is to say if I took this function here and plugged in a -2 I'd get ative -2 which gives you a positive2 okay so the absolute value function has to be two different functions one is that one because for positive inputs it's just yal X but for negative inputs it's yal negx and so yal X looks something like this 0 is 0 1 is 1 two is two cuz it's the do nothing function on the right now typically that do nothing function would continue on down here but that's not what happens what happens is all those values all those negatives become positive and so it goes up that way and creates a v function because all the negatives become positive and so it is a piecewise function If X is bigger than than 0 or equal to it's y = x if x is less than Z the function is actually y = x and so those are the two pieces that create that function few last things here in this section increasing decreasing or constant increasing and decreasing is very important Concept in calculus it really is visually very few struggle with this idea if you are walking from left to right on the graphs below just about anybody could tell me where you're walking uphill or downhill however in math we always talk about what is happening to the Y values the up or down values with the X values the left or right values this makes it a little more difficult but still reasonable so obviously the graph is increasing from here to here because you're walking uphill and so because you're walking uphill where are you walking uphill from well you're walking uphill from that value right there to to that value right there so from like - 1.8 to.5 you're walking uphill then you're walking downhill from here to here which ends up being from that blue marker that .5 to about 1.3 let's say it's right there so again that little stretch is uphill with the X values and this little stretch is downhill with y values and then obviously it goes back to uphill there from there kind of on to the right okay so where is it constant where where are you not going up or down well at the top of the hill or at the bottom of the hill that's where it is constant so look at this one again let's just do the X Parts because you can see where it's uphill and downhill so it's uphill from -2 pi to 3 piun 2 where else is it uphill from piun / 2 to posun / 2 where else is it uphill from 3 Pi / 2 to 2 pi okay where is it downhill from -3 pi over 2 to negative or actually all the way to negative pi/ 2 for those X values you're going downhill on the Y values and from Pi / 2 to 3 pi/ 2 it is also downhill the function so again you're talking about what's happening to the Y's with the XIs last one here where is it downhill well downhill is from4 to -3 downhill is from 1 and then obviously all the way to wherever that stops going downhill right there so one to about right there where is it uphill from neg3 to -1 and from um well that's it that's the only uphill so where's is it constant well it's constant at this point right there because it just first flat second it's flat and then from right there to right there it is also constant now if it asks you where is this function really constant yes those pink dots technically you're standing flat but there's no amount of distance you're actually traveling so really it's constant from negative 1 to one but you do have to know that at that pink dot right there it is neither increasing nor decreasing it is actually constant just for a split second all right that is it the homework is as follows paper pencil um show me all your work I grade the work not the answers um you will turn this in Via canvas you'll take a picture of each page and use an app to put all those pages together in the PDF which I will show you how to do all right good luck on the homework