So we're going to start today looking at working with functions. So this topic for most schools is the first bit of year 11. Some schools do do it in a bit of a different order but yeah it is important. do need your function stuff again in year 12. There's a fair bit to functions I would say, so we will jump straight into that. So the first thing I'm going to look at is what is a function?
So when you're talking about functions and when you do this at school, you're going to be introduced to two topics or two concepts. So the first concept is a relation and your second concept is a function. And so one of the key things you need to be able to do is to differentiate between a relation and a function. So a relation is a set of ordered points with coordinates x and y where your x and y variables are related according to a specific rule. So let's say you're talking about a function called a function.
So I've got here the example of x squared plus y squared equals 25. And I'm hoping that you recognize that that is the equation of a circle. And so essentially what that's saying is that whatever coordinates we put into that equation, they are going to fit a certain rule. So your x squared plus your y squared will equal to 25, right? Then we have a function, which is a specific type of relation, which... for only have one y value for any x value.
So essentially what that's saying is that when you have x values on your x-axis each of those x values is only going to be related to one single y value. So let's think about this in terms of the examples that I've got written down there. So with the example for a relation if you imagine a circle sitting on your coordinate axis let's say we've got you know you know, a point on your x-axis here, that x point is going to have a y point at the top and the bottom of your circle, right?
So with that, each of your x points has two y values, okay? So they are related. There is a rule that dictates that relation, but each of your x values has two y values. So therefore, it is not a function, right? Because the thing that differentiates a function from a relation...
is that your functions are only going to have one y value for each of your x values. Okay, so then looking at that example, y equals x squared, hopefully you recognize that as the equation of a parabola. If you're thinking about your parabola and you think about that in your head, any parabola can be positive, negative, whatever pleases you.
If you pick any x value on that coordinate plane, you're seeing it in your head, you'll recognize that each of those x values is only associated with one y value and that is what makes that a function okay if the x values only have one y value so another example of that would be a cubic function each of those x values only has one y value straight lines usually are only a functions unless they are vertical lines otherwise they are going to be be your y values as well okay with a function it's important to recognize that two x values can be associated with the same y value okay so essentially thinking about that parabola again parabolas are symmetrical okay so let's say if we have the function y equals x squared you're going to have a value of x equals one and at that point y is also going to equal one and then you're going to have what x is equal to negative 1, y will also be equal to 1. And that is still a function, right? So you can have the same y value for two different x values, so long as no single x value has more than one y coordinate. So yeah, that is how you tell the difference between a relation and a function.
We're going to look a little bit further about how we can tell the difference between relations and functions when we're looking at graphs. as well. So we do something called the vertical straight line test to test whether something is a function or whether it is a relation. Okay so something else that comes up, sorry I just remembered this, sometimes in exams is thinking about all functions are relations but not all relations are functions. Okay so if we go back to that slide.
slide before we'll see a function is a specific type of relation okay and the defining feature is that every x only has one y okay so a function is always going to be a relation but relations are not always functions. Okay, anyway, little side note there, we've got the vertical straight line test, and this is how you can tell whether something is a function or not. So for any graph you've got, this is... looking like the graph of sine and then you've got the graph of a circle over here. So for any graph you've got, if you draw a straight line through the graph at any point you should only intersect the graph once.
Okay so it's important to recognize that the straight line has to be drawn anywhere. There are some graphs where there'll be points where it only intersects once and other points where it intersects twice. So pretty much if at any point on the graph if you drew a straight line and you'd intersect the graph twice It's not a function.
Okay, so like this circle here We've drawn the straight line down this random point here and we can see that it intersects the graph twice so essentially what this is representing is that The graph has two y-coordinates for the specific x point because when we draw a straight line We are essentially drawing this equation a line with the equation x equals something. Okay and in doing that drawing a line with x equals something if it intersects the graph at two points that x has at least two y-coordinates. On the converse if we have a straight line going down our graph and it only intersects once that means that x equals whatever that line we've made it equal to only intersects the graph once therefore it only has one y-coordinate and that is how we can tell.
graphically whether something is a function or not. So you may be asked to do that in an exam to perform the straight line test or to determine whether a graph is a function or not. The other way is once you look at these enough you will be able to sort of just look at your graphs and pick up whether or not they're going to be functions by just looking is there any x coordinates that have more than one y coordinate. But that's not something that you need to do right right now that's just as you start to look more and more at functions you'll be able to start picking up whether something is going to be a function or not. So something important to know and you will use this from literally the beginning of year 11 to the end of year 12 is about function notation.
So you may have been introduced to this in year 10, some schools do introduce it, some schools don't. It doesn't matter. So if y is independent on the value of x then we say y is a function of x which is essentially written as y equals f of x. So this is a little bit more complicated. little bit tricky to think about.
So when we say we have f of x, so just this bit here, this is meaning x being the function, the numerical value or algebraic value that we are substituting into the function. Okay, so if we've got a function and it's x squared and we substitute x into that function, we are going to end up with x squared, right? If we have x... squared and we have f of 1 we put 1 into that equation we'll get 1 squared that's equal to 1. Okay so we'll have a bit more of a look at an example in a minute because it's a little bit tricky to get your head around without seeing an example but essentially what this means is that our y coordinate is equal to f of x so our y coordinate is equal to substituting x into the function. So since y is equal to f of x, then f of a is the value of y at the point of the function where your x is equal to a.
Okay, so whatever is in these brackets when we're looking at a function, that is what you're substituting into the equation in place of whatever pronumeral is in your function. Usually it is x. I mean, sometimes your equations are going to be given to you as like, you know, a or b or c or anything.
Usually it's x. they don't try and confuse you that much, but whatever value, so that can be a number, it can be a letter, is in this bracket, that's what we're substituting in, in place of our unknown in our original function. So we have come across our first example of example question for today. So if f of x is equal to x squared plus 4x plus 4 find f of a and then f of minus 3. So I will give you guys about 30 seconds or so just to have a go at this question.
If you need more time pause the video and then we'll go through it together. Alrighty, so if you need more time pause the video here because I'm about to do a big reveal of the answers. So, looking at the first part, so part A.
Our original function is fx. equals x squared plus 4x plus 4. When you have a function given to you like this, usually it is fx, essentially what that means is x is kind of just like your placeholder. So it's saying like we're not substituting anything in there yet, but when we do substitute things in, we're substituting them in. in place of this x.
So that means when we have f of a, we are putting a in wherever x was in our original function. So then we end up with a squared plus four a plus four, right, because all we've done is substitute a in place of x in that original function. So doing it with pronumers is typically pretty easy because you're essentially just exactly replacing your x's for whatever pronumer you've been given, okay? Doing it with numbers is a little bit trickier. So we'll have a look at this example for part B.
So we've got f of minus three. And what that is, is substituting in minus three, wherever there was an x in our original equation. So what we do is we do minus three squared.
Please remember when you're doing these questions, if you are being asked to substitute in a negative value, you must put brackets around them. Otherwise you're going to end up with the wrong answer. cell when you square it okay if you don't put brackets around that squared and you chuck in your calculator your calculator here would spit out negative nine but that's not the case here because we need to substitute this whole entity so this whole negative three where the x is and in this case the whole x is being squared so the whole negative three has to be squared right so you put it in the brackets square it and your calculator will spit out nine for you um depending in your school depending on the markers and stuff you get some markers will actually mark your working as incorrect if you don't have your brackets around your negatives in this sort of situation so do keep that in mind we've got negative three all squared plus four negative three plus four see in that first line all we've done is substitute in negative three wherever there was an x in our original equation then when you have actual numerical values perform the equation so you can either do some mental maths or pop in your calculator I would say it's always good to write out every step if you're working just in case something goes wrong so then you've got negative 3 all squared is equal to 9 4 times negative 3 is equal to negative 12 plus 4 and then once you add all of that up you'll get 1 so therefore we can say f of negative 3 is equal to 1 okay and so that is how we do simple substitutions using functions and essentially that's the best example for using function notation and what function notation is useful for.
So then the next important thing which tends to be most students least favorite part about functions is domain and range. Now speaking from my experience I didn't love domain and range but once you sort of get used to it it's not too too tricky. So You may have been introduced to this in year 10, so you may remember that your domain is just the x values for which a function is defined.
So on which x values are we going to have a matching y value? And then your range is kind of the opposite. So it's.
the y values for which a function is defined. So essentially where on the y-axis is there a matching x coordinate? So if we're looking at this example question here, find the domain and range of y equals x squared. first thing I can say to you is your domain and range questions are so so so so so much easier if you draw yourself a graph because then you can actually like literally see where things exist okay so we've got this graph here for y equals x squared. It is a pretty basic graph.
You probably can see it in your head as well. It's just the basic graph of your parabola. You haven't moved the vertex at all. So your vertex is here at 0, 0. So if we're thinking about the domain, we want to think about the fact that our parabola, unless we've put limits on it, will continually just go wider and wider and wider and wider and wider and wider along both sides of your x-axis. So essentially what we can say for the domain is that our domain is equal to all real x or x from negative infinity to infinity.
So the reason that we can say this is because our x's just get wider and wider and wider and wider. So for every single x value there will be a coordinating y value because we've not put any limits on how wide x can get. Then when we're looking at range we want to see with parabolas we look where is our vertex and our range is from the vertex up if you've got a concave up parabola if your parabola is concave down your range is from the vertex down so here we've got our vertex at 0 0 so then our range is going to be y greater than or equal to 0 or you may write that as 0 to infinity for your range for y so the reason for that is because for for every value of y greater than 0, there is going to be a corresponding value for x that gives a definite coordinate on our x-axis. Okay, so that's a pretty simple example of domain and range, so it will get a bit harder when you start doing more examples in class and things like that, but just as being your introduction to domain and range for today's little... bit on functions we'll just leave it at that.
Just remembering that your domain is for what values of X are you going to have a matching Y and your range is for what values of Y are you going to have a matching X. Okay and yeah if I can give you any tip for domain and range draw the graph it's so so much easier when you can see what you're trying to find the domain and range of. So we'll have a little bit of a look at hyperbolas.
So the next little bit of today's lecture is just going to be I guess a focus on some of your specific function types that you need to know of and that you need to be able to sort of use and manipulate. doing a whole bunch of different things, but we're going to look at each of them together. So the first function we're going to start with is your hyperbola.
Some schools and some textbooks will also refer to this as the graph of the reciprocal. So if you see see it written like that that just means hyperbola and so a hyperbola is a discontinuous function in the form x y equals k or y equals k on x where k is a constant value so essentially what that means here is that if we have a discontinuous function pretty much just means there's two segments of our function and that there's going to be some values for x or y with hyperbola it's talking about x but discontinuous can also mean y, where the graph doesn't exist. Okay, so there's going to be some values for x where our graph cannot and will not exist.
Now these equations may look a little familiar, xy equals k or y equals k on x, if you remember doing inverse variation. So these do look like the graph, I mean the equations for inverse variation as well. Okay, so yeah, discontinuous function means you...
function it's not a continuous curve there's at least two segments to it some graphs will have more and so when we're looking at a hyperbola specifically our graph is in the form of two discontinuous curves that have asymptotes in the x and the y axis that they cannot cross so an asymptote is essentially just a line which your curve will approach but it will never touch so looking at this graph of a hyperbola here we can see that this is just the basic graph of y equals 1 on x so that's like your hyperbola kind of like how if you had a parabola like your basic parabola is y equals x squared your basic hyperbola is y equals 1 on x and so essentially at that point your asymptotes exist at x equals 0 and y equals 0 and your graph will never hit those two points if you just have the basic hyperbola and then your asymptotes will change as you change your hyperbola so as you move it up down left, right, dilations, or anything like that, you will change your asymptotes as you change your function. So looking at some more specifics of our hyperbolas, if your K is positive, then your curve is going to exist in the first and the third quadrant. So I'll show you guys this example graph here again.
Remembering whenever you do maths, your quadrants are labeled. one two three and four okay starts in the top right goes anti-clockwise okay this come will come um back when we look at trig um we do some stuff with trig your quadrants and trig are also labeled as such one two three and four so if our value for k is positive so if we have a positive constant then we are going to have a graph in the first and the third quadrant if k is negative then our graph is going to exist in the second and the fourth quadrant. Then when we're looking at our equation again the larger our a is that means the further away the curve will bend from your asymptotes when we're looking at the equation in this format y equals k over x minus a plus b where a and b are the transformations you can do to your graph where it'll move it left and right up and down. Okay so similar to moving any other graph if we have X minus A this is moving our graph left and right and then the plus B is moving it up or down depending on the values for A and B. So the bigger A is the further away the curve is bending from the asymptotes.
Okay when we have our graph in this format our asymptotes our x asymptote is equal to a and our y asymptote is equal to b. Now the reason for this is because we can never have our bottom of our denominator, bottom of our fraction or also known as the denominator can never equal zero. So x minus a can never equal zero so therefore x can never equal a.
The reason that b becomes the y asymptote is because our original asymptote occurs at y equals zero. So if we move it up or down, we're essentially just moving that original asymptote with us. Okay, so another example question for you guys to have a go at. I'll give you maybe a minute or so.
Have a look at this question. See if you'll be able to graph y equals five over x minus two minus three. And then we'll go through it in about a minute.
If you need more time, you can... can just pause the video as well. So I'll just give you a little bit a minute to have a go and look at this question.
Alrighty, so looking at this example question, first thing we want to do is rearrange our function so that we can work out where our asymptotes are going to be at. So we can see that our asymptote here is going to be minus is going to be 2. So our a value here, and then our y asymptote is going to be at negative 3 because in our original function, we've got our asymptote at negative 3. I'm sorry, I'll value for b at negative three, which means the asymptote is then going to be at negative three. So from there, what we want to do is find out intercepts.
Now, hopefully by now, you've had a lot of experience finding intercepts. So we find our x-intercept when y equals 0 and our y-intercept when x equals 0. So that's going to give us these two coordinates as our x and y-intercepts. So once we've found the asymptotes and the intercepts, we're ready to sketch our graph.
Okay, so hopefully you guys have had a go sketching. So we end up here with this graph. So the first thing we do when we sketch is always plot in your asymptotes. So we're going to do this. draw them as, or they're usually drawn as, and how I like to draw them is just as dotted lines.
Make sure you use a ruler so that they're accurate, and we want to make sure that they are to scale. So the first thing you want to do, draw in your asymptotes. Second, we want to be working out which quadrants is our graph going to exist in.
So remembering that your quadrants are going to be relative to your asymptotes, okay? So if we're looking at these green lines here, oh. this creates our new four quadrants. So we've got 1, 2, 3, and 4. Okay, so all of this, even though some of it is above the x-axis, some of it below, all of this makes up quadrant 1, all of this quadrant 2, quadrant 3, and quadrant 4. So going back to our original equation, we can see that we've got a positive value for k.
So our graph exists in the first and the third quadrants. Then we just want to plot in our intercepts. So So our x-intercept, our y-intercept, and then we just want to draw our lines for our hyperbolas going towards both of our two asymptotes. Just like that for hyperbolas.
So hopefully that made sense. And yeah, we're going to have a go now looking at circles. So circles have a very distinct equation and you'll be able to pick up the equation.
of a circle really easily. They exist in the form x squared plus y squared equals r squared or alternatively as x minus a squared plus y minus b squared equals r squared. So this is your basic equation of a circle if you've got x squared plus y squared equals r squared. And the center is 0, 0 and the radius is equal to r. So in general The domain and range of this basic function, the domain is between r, x, and negative r, and the range is r, y, and negative r.
Okay, so essentially the reason that is, is because your radius spans the widest point on your x-axis on both sides. You'll have r and negative r as the widest points, and then y and negative y as your shortest. and total lost points.
Then the equation of a circle with the center a b and radius r is given in this format. So you'll have x minus a squared plus y minus b squared equals r squared. Okay so essentially when we're looking at this as a more complex equation for a circle that's had some movement to it we can find the radius of another radius the center of our circle with a and b. It's important to notice here that minus a and minus b give you the coordinates a and b for your center.
If you had x plus a and y plus b, you'd have the coordinates negative a and negative b. So the center is the opposites of what we've got going on here. And then again, the same thing, your radius is the square root of whatever all of that's equal to.
Okay, so next example. question I want you guys to have a go sketching this circle and stating its domain and range if you've got x minus 3 all squared plus y plus 1 all squared equals 4. So I'll give you guys probably about one minute to have a go at this and then we'll look at the answers together. Again, just as a side note as you guys are doing this, if you are watching live, feel free to ask any questions.
Because I'll be sitting in the chat to answer your questions as well as we're going on. And feel free to pop your answers in the chat as well. So if you can find the domain and range, pop it in the chat as we're going on as well. So I'll give you guys maybe 20 more seconds just to have a go at this.
Alrighty, so let's have a look at this question together. So again, finding domain range is so so so much easier when you have a picture in front of you, because you'll actually be able to see, specifically with circles, they do end. really distinct points. So a bit of a tip there, your domain and your range for your circles is never gonna be to infinity because your circle is a defined little thing. So this is what we end up.
with as our circle. So we have center here at 3 minus 1 and we can see that because that's the opposites of our a and our b is 3 and minus 1. Then we've got 4 here so that's r squared so therefore r is equal to 2. So when you're drawing your circles plot your center and then go 2 in each direction and then just draw the curve around your circle. When you are drawing circles in your exams your examiners are not looking for you to draw perfect circles they acknowledge that it's hard to get the curve exactly right but they want to see that you've got your center right even what they want to see you've got these four distinct points and then they just want to see sort of like a general you've tried to draw a curve connecting them they're not looking for exact perfect circles.
So We've got here five and negative one, oh sorry, and positive one as our domain. So we can see that from our graph. Otherwise, what we can do is think if our center coordinate for x is three, we just plus two and minus two on either side, which gives us the domain of x is between five and one. And then we can see from the graph that our range is from one to negative three.
Otherwise, our y coordinate is negative one so we can just plus two and minus two to work out our range here as well. Okay, and then that's all you really have to do when you have to graph a circle. So in some cases, you're going to be given the equation of a circle with x squared plus y squared plus ax plus by plus c equals zero.
And you're going to be asked to solve this. Sometimes if whoever's reading your paper is nice, they'll tell you that you have to solve it by completing the square. Otherwise, they'll just give you that equation and you're sort of just expected to know that you need to come back.
complete the square. So the way that you're going to recognize this is because it'll be in your exam as x squared plus y squared. There may or may not be ax and by but there will always definitely be x squared and y squared and that's how you'll recognize that you've got the expanded version of the equation of a circle and you're going to need to complete the square in order to find it.
So that equation when you've got x squared plus y squared plus ax plus by plus c is essentially just the expanded version of the equation of a circle with a few of your factors missing which is essentially what you're trying to create with completing the square. So by completing the square you find the missing factors so you can then find the whole equation of the circle. Now completing the square Something that you do look at in year 10, which is why I don't have an example of it just yet.
So what I'm going to get you guys to do is have a go completing the square for A and for B. I'll give you probably about a minute or so. See if you can remember it. If you don't, we're going to go through it together anyway.
So see what you remember about completing the square and then we'll go through the example together. Okay, so give me about 10 more seconds maybe Again, if you need more time just pause the video so you guys can complete it or if you've had enough of waiting and you're not sure how to do it you can skip ahead and Look at the solutions so When we are complete the square, what we do, we've got x squared minus 8x, we half this, we half our coefficient of x, and square it, okay, so half of negative 8 is negative 4, Negative 4 all squared is 16. Okay? So we do the plus 16. Then you've got y squared plus 10y.
Half the coefficient of y equals 5. Square that is 25. Now, what we've done on this side is we have moved our constant of 32 across. And then we've added the 25 and the 16 to this side as well. Okay?
Because essentially we can't just go adding, like, numbers into the equation. What we have to do is make it... equal on both sides.
So as long as we add 16 and 25 on both sides our equation is still equal. So then we add the 25 we add the 16 and we end up with this equation here. Now with the two examples that I've given you I actually have given them to you in order so I've got x squared minus 8x plus y squared plus 10y.
Sometimes it'll come a bit muddled so it might be like x squared plus 10y plus y squared minus 8x. First thing you want to do, arrange it so you've got your x's together, your y's together. Then half your coefficients and square them, making sure you add them to both sides. And then you'll get something looking like this.
Now, from there, what you can do is go x minus 4. So in this case, x and whatever you halved. So the half of the minus 8 gives us minus 4, and we square that. Okay. k all in brackets. Same thing with the y's you do your y plus whatever we halved which was plus five all squared and then that nine comes from just doing that maths though.
And then you can see here that you have the equation of a circle so we can say our centre is going to be at four negative five because we're just taking the opposites and our radius is three because nine is representative of r squared so you get our radius of three. Okay, so then looking at the second equation, if we had x squared minus 2x plus y squared plus 24y plus 120, again, recognizing that this has sort of been sorted out for you, all your x's are already together, the y's are already together. So what we want to do is we've got x squared minus 2x, half of 2x is minus 1, minus 1 squared, we just plus 1, plus 1. Then same thing with the y's, we've got y squared plus 24y, half of 24 is 12, 12 all squared is 144, so we add the 144 to this side as well.
From there we can start creating our sets of brackets, so we open our first set of brackets with x, just our per-numeral, whatever we halved, all squared, then y plus whatever we squared, which is 12, all squared, and then we do this math for 120, negative 120 plus 144 plus 1, we'll get 25. from there we use our opposites to find the center of our circle so here we're going to have a center of 1 negative 12 and our radius is going to be 5 because 25 represents r squared So therefore our r is equal to 5. And that's pretty much all you need to know how to do with your circle. So then this is looking at semicircles. So the equation of a semicircle that's above the x-axis with center 0, 0 and radius r is going to be equal to y equals the square root of r squared minus x squared.
So that will have domain negative r and r and range between r and 0. So that's when we have have the center as well so keep that in mind. And then when we have the equation of a semicircle that it goes below the x-axis again with the center and the radius r we're just going to put a negative in front of that square root sign and that means it's going to be below the x-axis as well so it's being going to be the half of the circle that it goes below the x-axis that's going to be concave up. yeah, that's pretty much all we really need to know about semicircles for now.
They're a little bit confusing when you start to move them around a little bit, so we won't go into that today. I'll leave that for you guys to go into in class when you do functions, but just sort of knowing that in general this is what the equations of semicircles looks like is going to be enough for you guys for now. So next we're just gonna have a bit of a look at absolute values. So you may know absolute values but you might not know what the graph of an absolute value looks like but essentially the absolute value is just the positive distance away from zero denoted by these sort of straight lines with x or whatever number in the middle essentially you're just removing the negatives if they're there Yeah, so it's pretty much the distance away from zero that something is Disregarding the fact that it's moving in the positive or the negative direction.
It's just like the absolute positive distance away from zero so through these. So essentially, the absolute value of negative three is just three, because essentially, we're just removing the negative, the absolute value value of three is just three, absolute value of negative 11 is 11, absolute value of 11 is also 11. Then we have an equation here using some absolute values to solve. So 5 minus the absolute value of 2 minus 3. So this is equal to negative 1. So then the absolute value is going to be 1. So we've got 5 minus 1. So that's 4. And then we've got 4 plus the absolute value of 5 minus 8. So 5 minus 8 is going to be equal to negative 3. The absolute value of that is just going to be equal to 3. So then 4 plus 3 equals 7 minus the absolute value of 5. So 7 minus 3 equals 7. minus 5 is going to give us 2 and that's pretty much how you use the absolute value and then there's also a graph that helps us denote the absolute value so we know that absolute values make things positive so therefore when we're given a graph that's an absolute value every single y value is going to be positive so essentially what you're doing is just making all of your y values positive and your graph ends up looking like a So when you're sketching your absolute value graphs, all you really need to do is sketch one side of them and then they're symmetrical, so you just draw the exact same thing on the other side. And all your absolute value graphs are going to look like a V. You move this graph around the same way you'd move a parabola or a straight line, you know, you can do the plus or minus, to move left and right, up and down as well.
Changing the gradient. makes it steeper or wider as well just like it would for a regular graph of a straight line. Alrighty and the last little bit of functions we're going to look at for today is looking at even and odd functions.
So even functions are functions that are going to be symmetrical about the y-axis or in other words they're going to have a line of symmetry about the y-axis. So this is seen in things like semicircles and parabolas, and that they are exactly symmetrical on either side of your y-axis. So this can also be true for that you move around, so they're not always going to be symmetrical about the y-axis, but if you move your parabola to the left or to the right, it's just going to be symmetrical about the middle point at the vertex.
Okay, and so the reason that we can prove this, or how we prove this, is if you can prove that f of x is equal to f of negative x you're going to have an even function okay so the easiest way to think about this is thinking about it in terms of y equals x squared so i'll just our normal fx is equal to x squared and then if we have f of negative x we're going to have negative x all squared and we know that if we do a negative squared we just end up with positive so it's also going to be equal to x squared okay then the opposite is the odd function and so odd function are functions that have a point of symmetry about the origin. So essentially what that means is if your graph is rotated 180 degrees about the origin, it's going to give the original graph as well. So this is seen in cubics and hyperbolas and so essentially what that means is if we rotate this graph 180 degrees around, if I line it up properly, we're going to have the exact same graph.
Okay let's rotate it a hundred degrees. 180 degrees, exact same graph. Okay, 180 degrees, the exact same graph. Okay, so essentially that's what we mean when we're looking at odd functions.
Okay, so you can prove something is an odd function if f of x is equal to negative f of negative x. Okay, so essentially if we look at this graph here, we have y equals x cubed as our fx, and then if we had negative f of negative x you'd have negative of the negative x cubed which you'd end up with negative times the negative which would get you back to just x cubed. Okay so we've got some example questions to look over so I'll give you guys maybe a minute or two to have a go at these questions to prove that some of these are even, odd or neither. possible that your graph is not going to be even and it's not going to be odd. So yeah, I'll let you guys have a go at these.
I'll give you about two minutes. Type your answers in the chat if you're watching it live. If you want more time, pause it.
If you don't want any time, skip ahead to the solutions. Okay, so we'll go through the solutions for part a first. So prove that f x equals x squared minus 4 is an even function.
So we know that something is going to be an even function if f of x equals f of negative x. So we need to find what is f of negative x and we do that by substituting in negative x wherever there was an x in our original equation. So then here we're going to end up with negative x all squared minus 4. Once we do this, this square we know that if we square a negative we just end up with the positive so we're gonna get x squared minus 4 which is equal to our original fx so therefore we can say f of X is an even function and it's as simple as that to prove that something is even. To prove that something is an odd function if we've got fx equals negative f of negative x. Okay so for our original function here x cubed for part B if we want to find negative f of negative we wanna do negative of negative x cubed, which we then end up with two negatives that when we multiply them together, we'll cancel each other out to get just x cubed, which is equal to f of x.
So therefore our fx equals x cubed is going to be an odd function. because f equals negative f of negative x. Okay and that's all we have to do for that one.
So then for part c, prove that f equals 3x plus 5 is not even nor odd. We first need to disprove that it's not even. So whenever you get a question that says, prove this is not even and prove it's not odd, separate into two sections, first prove it's not even, then prove it's not odd. Okay, so for something to be an even function, f of x has to be equal to f of negative x.
So in this case, f of negative x, if we substitute that in, we'll get 3 negative x plus 5, which leaves us with negative 3x plus 5, and that does not equal f of x. So therefore, we can say that our fx is not an even function. Then for proving it's not an odd function, an odd function occurs if f of x equals negative f negative x.
So in this case, to find negative f of negative x, first substitute in negative x to where there's x, and then we need to make all... all of that entire function negative. So then we end up with, when we multiply this out, negative three X plus five, all of that is negative.
So then we multiply that negative out, you'll get three X minus five, which does not equal out F of X. So therefore, F X is not an odd function. And we can therefore say that we do not have an even nor an odd function when we are looking at three X plus five.
Okay. Okay and that's all you need to do with functions in proving that they're even and odd. It can be a little bit tricky to get your head around the f of x and the f of negative x and the negative f of negative x, so I just recommend doing lots of practice with it, but that's the only two rules that you need to know to have to prove whether something is even or odd.
Alrighty so that brings us to the end of our overview of functions today.