CIE A LEVEL Maths FULL SUMMARY Topic by Topic Paper 1: Functions

Hey guys, just to let you know this lesson is in collaboration with Flip Academy, a free online platform for all of your GCSE, A-Level IEB and CBSE revision, out now on web at www.flip.academy and also as an app for Apple, iOS and Android. It really is better than all the other GCSE platforms out there, it covers all the topics in great depth and has an app that you can download your courses on and use offline. Welcome back to Flip Academy, today we're going to be looking at the Functions chapter of the CIE A-Level Maths spec, so this is going to be more Paper 1 stuff. So, first things first, we're going to go through a couple of definitions that you're going to be using during the course of the exam. So, you need to understand the terms Function, Domain, Range, One-to-One, Inverse Function and Compositional Functions. So, first off, what is a Function? Well, a Function is something that takes an input and it gives an output, right? So, pretty much everything we've dealt with in terms of equations and formulas are Functions, so whenever you see something like y equals and then something in terms of x, we call it f of x, so that could be 2x plus 3 or it could be x squared or anything like that, that is a Function, okay? It takes values for x and it gives you values for y, okay? Now, the question is, what can we put into the formulas? Well, this is called the Domain of a Function, so it's all the values that can be placed into a Function, okay? So, basically, in simple English, it is just the x values, simple as. Now, you might be thinking, well, isn't that just all numbers, right? And when we talk about things like x squared and 2x plus 3, then you'd be right. x can be anything, so the Domain would be written as x is any real number, okay? However, what if you had something like y equals 1 over x? Well, now, x can't be any real number, right? Because if I put in x equals 0, what happens to the Function? Well, what's 1 divided by 0? Well, it's a math error, right? It's undefined. So, in this case, the Domain would be x just simply doesn't equal 0. It's any other number except 0, okay? Now, the Range is just all of the outputs, okay? It's just all the values that y can take, okay? So, again, all the values you can put in for x is the Domain, and all the values that you can get out of a Function is called the Range. And we're going to do a few examples a bit later on about calculating what the Domain and Range is. It's not always just any random number. So, for example, if we took y equals x squared, what is the range of values that we can get out of it? Well, it isn't an infinite number, right? Because what is the smallest value we can get out? Well, the smallest this could possibly be, well, let's think. If we sub in x equals 0, then we get y equals 0, right? And if you put in y equals some positive number, you square it and you get a bigger positive number. What about when we put in x as a negative number? Well, when we square it, it becomes positive. The smallest this thing can be is 0. So, this is how we'd write the Range. You may also see it written like this. But it does functionally mean the same thing. y and f of x can be interchangeable. So, your Domain will be x is either any real number, or x isn't a certain value, or it's greater than a certain value, or whatever. And your Range will be y, or f of x is greater than or equal to 0, or it's any real number, or something like that. Now, the next definition is a one-to-one function. So, this is where things can get slightly complicated. So, what this refers to is x and y. So, when it's one-to-one, one x value gives one y value. So, for example, again, if we look at y equals 2x plus 3 again, if I sub in x equals 0, I get some y value. Sub in x equals 1, I get a different one, and so on. For every value of x, I will get a different value of y. Now, there's also many-to-one functions. And this refers to multiple x values give one y value. So, there are actually a lot of good examples for this. So, for example, with x squared, if I sub in x is minus 1, I get y equals 1. And if I sub in x equals 1, I also get y equals 1. So, there are two values of x which give the same value for y. The other examples are, for example, the trig functions, right? Sine of x. There are an infinite number of x values that give sine equals 1, for example, right? So, x equals 90, and then 180, and so forth. Or 360, and things like that. So, sine of 0 is 0, then sine of 360 is 0, sine of 180 is 0, and then so on and so forth. So, you can have many x values giving one y value. Now, there is one that's actually kind of illegal. You're not allowed to use it. It doesn't come up. As in, you can't have functions that have this. And that is a one-to-many. And that is literally just one x value can give multiple y values. Now, if you think about why this is illegal, what it means is, if I put into a calculator a certain calculation, every time I press equals, I can get a different answer. So, could you imagine if I put in, let's say, x is 1 here, and the values I can get out will change every single time. Again, that's impossible. You can't have that. When you put in a value, you will get one answer. However, you can get multiple answers for different x values. That's okay. But you can't put in one value of x and get multiple values of y. That will break everything. So, hopefully that all makes sense. Again, these are just definitions. You don't really need to know too much about them. But the next few things are things you're going to have to calculate. So, first of all is an inverse function. So, let's say I have, again, let's pick y equals 2x plus 3. So, I put in x values and I get out y values. So, something that you might think is quite useful is, what if I want to know what y value I'll put in to get x values? So, for example, in this case, we put in x and we get out y. But what if I know what y is and I want to work out x? Now, you might be thinking, well, that's just a simple rearranging, right? Very simple, very easy. But what you're doing there is actually finding the inverse function. You're finding what x values I have to put in to get a certain y value. So, an inverse function simply is finding the inverse, right? So, what you do is you find x values that give y values. So, for example, in this case, y, you double x and then plus 3. So, if you knew what y is, then you would just subtract 3 and half it to find out what x should be, right? You're doing the inverse, the opposite. And the last thing is compositions of functions. So, this is where I'm going to have to use f of x instead, right? So, let's say we have f of x. I want to make it nice and easy. It's 2x, right? And then I have another function, I'm going to call it g of x, is equal to, let's say, x plus 4. So, what I can do is I can sub in a value into f of x and use that value and sub it into g of x. So, the notation for this could be something like this. So, what I'm doing is I'm taking the entirety of the f of x function and subbing it into the g of x function. So, that would give me 2x plus 4, which is going to be just 2x plus 4. Now, of course, the functions that you're going to be dealing with are a bit more complicated than that, but all it is is a composite function or a composition of functions is just taking one function and subbing it in as x into another function, which is exactly what I did here. You could also do it the other way around, right? We could also have f g of x. Where I'm taking the g of x function and I'm subbing it into f of x. So, all I would do is I write the 2 and now x is now equal to x plus 4 and that would be my answer. You'd probably expand it. So, a lot of definitions might be slightly tedious, but hopefully that all makes sense. Let's get on to some exam questions. So, first things first, we have that identifying the range of a given function in simple cases and find the composition of two different functions. Determine whether or not a function is one-to-one and find the inverse of one-to-one functions in simple cases. So, one key thing to remember is, first of all, find the range of x. How can we find the range of this function? Well, it's a quadratic. Now, if we want to find the smallest value a quadratic can be, we use something called completing the square. And completing the square is, let's say you have something like x squared plus bx plus c. Completing the square simply says x plus b over 2 squared minus b over 2 squared plus c. And what this will give you is the minimum point, so the smallest value that our function can give you. Now, this is really useful because remember what the range is, it's all of the values our function can give us. So, if we find the minimum, then the answer will just be f of x is greater than or equal to whatever the minimum is. Makes sense, right? So, let's sub in these values. We have x and then b is 4, so 4 divided by 2 is 2 squared, minus 2 squared plus 1. So, we have x plus 2 squared minus 4 plus 1. So, we have x plus 2. And we'll label this part a. x plus 2 squared minus 3. So, the min point is just going to be minus 2 minus 3. So, remember this is the y coordinate, this is the x coordinate, so this is what our f of x is going to be. So, now we know that our range is f of x is greater than or equal to minus 3 because the lowest value that y or f of x can take is minus 3 because that's what the minimum point is. Not bad. Now, part b, it might slightly contradict what happens later, but we'll show it. It says, explain why the function f of x does not have an inverse. So, whenever you have a quadratic, it is a many-to-one function, always. And, unfortunately, many-to-one have no inverse. Technically speaking, many-to-one functions do not have a valid inverse for good reason. So, all we have to do is write many-to-one function has no inverse. So, whenever you see a question like this, explain why it doesn't have an inverse, all you have to do is say that it's a many-to-one function. And, again, that's because we have a quadratic, we have a squared. Now, whenever you have a squared, it means you're going to have two values of x that will give any particular value of y or f of x in this case. So, again, this is going to contradict a slightly later question, but that's it. Keep in mind, they can ask you to work out the inverse of a function, even if it's many-to-one. So, just keep that in mind, but it isn't actually technically a valid inverse function. So, illustrate in graphical terms the relation between a one-to-one function and its inverse. So, they give you a function here, we have to state the range, sketch the graphs of the f of x and its inverse, just like it says in the spec point, and then find an expression. So, let's get to it. Stating the range is not too bad. All we need to do is factorise this. So, you can either factorise it or you can find the minimum point. Both will give you the same answer. So, let's use completing the square because that's what I said before. So, f of x equals x plus 2 divided by 2 is 1, minus 1 squared plus 1, 1 squared minus 1 plus 1, and that's it. So, in this case, the minimum point is just minus 1, 0. So, our range is going to be f of x is greater than or equal to 0. So, let's get started. Now we need to sketch the graphs of f of x and f minus 1 of x. So, this here is how you denote an inverse function. And there's a few ways to do this. So, what I'm going to do is I'm going to start with just drawing the regular old f of x graph. So, let's get to it. No, I want to redo that. There we are. Now, minus 1, 0 is going to be our y-intercept, right? So, what we're going to do is draw that here. Let's make it this. And we're going to draw a nice, relatively okay curve. Now, I need some of the y-interceptors. So, the best way to find a y-intercept is to sub in x equals 0, which, as we can see from our equation, 0 squared is 0 plus 2 times 0, which is 0, plus 1. So, y equals 1. So, it crosses the y-axis at y equals 1. Okay. Now, there are two ways to talk about drawing the inverse function, okay? The first way is to know that whenever an inverse function is just equal to... So, inverse is equal to a reflection in the y equals x line. So, if you like reflections, if you're one of those people, you're more than welcome to do it. So, you will need to use this either way to find out what the shape is. However, there is a little hint I can give you as well. When you take an inverse function, if you remember what I said, right, a regular function takes in x values and gives you y values. An inverse function takes in y values and gives you x values. So, all you need to do is flip the coordinates. So, this point over here is minus 1, comma, 0. Now, when I do the inverse, it's just going to become 0, minus 1. I swap the positions of the coordinates because the input and output have swapped. And likewise, 0, comma, 1 for the y-intercept when I do f of minus 1 of x now becomes 1, comma, 0. So, we can now plot those points. So, 0, minus 1, which is here. And we have 1, 0 is here. Now, this can help you decide how your graph looks. But again, remember, it has to be a reflection here. So, we're kind of flipping it on its side. So, it should look something, and bear with me, something similar to that. So, keep in mind that you don't have to be an artist to do this. What they're looking for is, can you see this looks kind of like a reflection, very vaguely, kind of, maybe. The oval shape is correct. The coordinates are correct. So, you would get full marks for this. So, again, it doesn't need to be perfect, but the idea is that you do need to have the general shape. This should technically be a bit narrower, but I would get full marks for that. Now, for part c, the illegal question. Find an expression for the f of minus 1 of x and state its domain. Now, again, this inverse function shouldn't exist, but we're going to do it anyway. So, the way I find inverse functions is I'm going to write y equals x squared plus 2x plus 1. Now, here's how I do these. Step one, rearrange for x. So, what I want to do is get x equals. Now, you might think that's pretty tricky because it's a quadratic, but have a look at this. We're going to factorise it just like we had before. Now, what we're going to do, and this is where the illegal part comes in, is we're going to square root both sides. So, we get square root y equals x plus 1. So, the illegal part here is that whenever you square root, you should get a plus minus. However, when we do the inverse function, we can't exactly have a plus minus in our equation. It's not the most obvious thing. But, for the purposes of this question, we're going to leave it and subtract 1. Okay, there you go. Now, the second thing is all you do is you replace x with f minus 1 of x and y becomes x. So, I have f minus 1 of x equals the square root of x minus 1. And that's it. That is how you work out an inverse function. Now, the next thing is write down the domain. So, you might think, well, hang on a minute, I can do anything I want in that question, right? Not quite. Although it looks like you can sub in anything that isn't negative, that's one way to look at it. So, if I look at the square root, sub in anything that's greater than or equal to 0, that's one way. The other thing you can do is you can know that when you do the inverse, so again, draw a little line, the domain becomes the range and the range becomes the domain. Because again, remember, domain is all of the x values that I can sub into a formula. So, when I'm taking the inverse, that now becomes the range, all of the outputs instead. So, you know when you said, like with a normal thing, range is y, domain is x. But when I do an inverse function, I want to find values of x depending on the value of y. So, now the new thing flips. The domain becomes the range and the range becomes the domain. So, if I want to find the domain of this function, I can look at the range of the other function, which is x is greater than or equal to 0. So, in this particular question, you could have looked at this and decided, okay, x is greater than or equal to 0. Good job. However, sometimes, just looking at the formula doesn't always give you the answer with inverse functions. With inverse functions, you sometimes have to use this rule over here. Sometimes. And the reason for that is that sometimes the only reason you have a limit on your range or domain is because the original has a limit. It has a limitation. So, keep that in mind. In this case, it worked out fine either way you did it. In some cases, there might be a restriction. That's because of the original function. So, for example, imagine this. Now, imagine if x had to be between 0 and 5. For argument's sake. They can put that on there, which means the range of this now becomes f of x is in between 0 and 5. Even though there's no reason for the limit, as in just by looking at it, because the original has a limit, the inverse function will also have that same limitation. So, next, understand and use the transformations of the graph y equals f of x given by the classic f of x is plus a, f of x plus a, and a f of x and f a of x and simple combinations of these. So, these are all ones you've learnt at GCSE. So, this one here, all it means is you would add a to y coordinates. All this will be is you would subtract a from x coordinates. Remember, if it's inside the brackets, it affects x and it does the opposite of what it says. If it's outside the brackets, it affects y and it does exactly what it says. This one's outside the brackets, so it multiplies y by a. This one's inside the brackets, so it divides x by a. However, this is A level. They have added a new type of transformation they can give you, and that is the modulus. Now, with part a, not too bad. What we're going to do is, and it says on separate diagrams, so that's quite helpful. We don't have to squishy-squish everything up. So, we have 2 f of x minus 1. So, we are given two coordinates, a and b. Those are the only ones we have to include. So, for the three marks, you get one mark for each coordinate you get correct, and you get one mark for the shape. So, one mark for a, one mark for b, one mark for the shape. You don't have to include the minimums, maximums, x or y intercepts, because you can't really work them out anyway. So, 2 f of x minus 1. We're going to double y and then subtract 1 from it. And that's it. No other difference. It doesn't affect x. So, with point a, double that is minus 8, minus 1 make it minus 9. So, what we're going to do is, we're going to draw, and I'll extend this a little bit. This will be 3 minus 9. And with b, we're going to double that, makes it 10, minus 1 makes that 9. So, then we're going to have 7. So, you may be wondering about the shape. Well, with things like this, it should technically be squished together a bit, but they're not too fussed about it. So, they're only worried in bigger changes. But let's make it look a little steeper, so we can make it look a little bit more sharp. Now, b, this is the interesting one. So, technically I should label these x and y, so let's do that as well. Now, these symbols here means make positive. So, for example, if I had, all that does is it makes it plus 3. And then if I already had a plus 3, it just keeps it as plus 3. So, what this is telling me, that all my y values, if they're negative, I'm going to make them positive. So, what that means is, if you look here, everything below the x-axis is going to be reflected up. So, what this is going to do is it's going to basically make, the way I like to think of it, it's going to make everything below here bounce off of the x-axis. So, instead of going down here, that point A is going to be 3,4 instead of 3,-4, because all the negatives become positive. So, it's going to bounce off, it's going to be quite sharp like this, and go back down, and then bounce up again, and then don't forget the bit on the end. So, that's what the overall graph will look like. It'll come down here, bounce up, bounce up, bounce. So, everything below the x-axis is reflected, not everything in general, because everything is being made positive, y value in this case. So, all of the y values are made positive. So, just bounce, bounce, bounce. And that brings us to the end of the functions topic for Paper 1 CIE A-Level Maths.