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Hi, I'm Mrs. Dennett, and in this lesson, we're going to be looking at composite functions.

So what does the word composite actually mean? What are composite functions? Well, composite means that something is made up of two or more parts of something else.

So here we've got some functions, and they're going to be acting together.

So we've got a function f of x and a function g of x.

And what part a is asking us to do is take the function g of x, substitute -4 into it, and then whatever answer we get, we're going to put into the function f of x.

So the order is very important here, so we're going to be doing the g of x function first 'cause that's the one that's immediately next to the brackets, and then the function f of x is going to act on the answer, okay? So let's start by substituting -4 into our function g of x.

So we get 2 times -4, add 5, which gives us -3.

But we haven't finished yet because we need to put that -3 into the function of x now.

So we get -3 multiplied by 7, and that is -21.

So there we've got the function g of x and then the function f of x acting on our answer.

In part b, we've got function acting on a function again, but this time it's the same function.

So we've got two that we want to substitute into f of x, and then after we've done that, we want to substitute that answer into our function of x again.

So it's almost acting on it twice.

Well, it is acting on it twice.

So we're going to do 7 times 2, which gives us 14.

So we've applied the function once, and then we're going to apply that function to 14 again.

So we put 14 into 7 times x or 7 times 14, and we get an answer of 98.

Here are some questions for you to try.

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Here are the answers.

Be very careful about the order in which you apply the functions.

Start with the function closest to the bracket first.

We're now going to look at some questions that have got some more complex functions in there.

So this time, we're going to work out gf of four.

So again, we want the put four into out function of x first.

Remember, the order is really important.

And after that, then we'll put the answer that we get to function of four into our function g of x.

So let's put four into f of x, and we get four plus three equals seven.

And then we put 7 into g of x, so we get 7 squared minus 5 equals 44, and that's our answer to that composite function.

Next, we're going to look at applying g of x twice to x.

So we want to substitute x into g of x, and that's pretty straightforward because we just get x squared minus five.

And then we're going to put this answer, x squared minus five, into g of x again.

So this time, we've got g of x squared minus five, so we're going to get x minus five all squared and then take away five.

So we expand that bracket.

I like to write it out twice so I don't make any mistakes or miss any terms out.

Expand it to get x squared minus 10x plus 25, and don't forget to take away that 5 on the end.

It's really important.

It's a common mistake that people will forget about taking away that five at the end if you've got an extra term at the end of your function.

So we simplify that, and we get x squared minus 10x plus 20.

Here are some questions for you to try.

Pause the video to complete the task, and restart when you've finished.

Here are the answers.

The last three questions are a little bit trickier 'cause you're substituting in more the one term into the second function.

I find it useful to put brackets around the terms and substituting in to help me.

Here are some more questions for you to try.

Pause the video to complete the task, and restart when you are finished.

Here are the answers.

The last question involved expanding x squared minus 7, all squared, which gives x to the power of 4, take away 14x squared, plus 49, and then take away the 7 to get 42 at the end.

We're now going to look at what happens when we've got three functions.

So in part a, we've got a function h of x that is then being acted upon by g of x and finally f of x.

So fgh of x is what we say.

So the first thing we do is we apply h of x to x.

So we substitute x into h of x, and that's quite straightforward because we're just substituting x in, so we get three divided by x.

We now want to take this answer and apply g of x to it.

So we've got g of three divided by x, and that would give us three over x, all squared, take away five.

So if we simplify that, we get nine over x squared take away five.

And finally, we want to apply f of x to this answer, so we substitute in nine over x squared take away five into the function of x, and we get nine over x squared take away five plus three.

Don't forget to add the three on the end there.

Very important.

And when we simplify that, we get nine over x squared take away two.

For part b, we want fg of x added to gf of x.

And be very careful here 'cause these don't give the answer answers, and you'll see this shortly.

So we start with the first part of the addition fg of x, and we apply the function g of x.

So we substitute x into x squared minus five, and that just gives us x squared minus five.

And then we apply the function f to this answer.

So we substitute x squared minus five into the function of x, and we get x squared minus five plus three.

And we can simplify that to x squared minus two.

Then we need to look at gf of x so that we can add it onto this first function of a function that we've found.

So we look for f of x first.

So we substitute f of x into there, and we get x plus three.

And then we apply g of x.

So we put x plus three into the function g of x, and we get x plus three all squared.

Take away five.

And we expand that to get x squared plus 6x plus 4, and then we need to add together these two answers.

Now, notice, we got different answers because we were applying the functions in different orders.

There are some cases where we do get the same answer, but not always, so you have to be really, really careful not to just assume that you'll get the same answer because you're applying the same functions to x.

So let's add these together.

So we get x squared minus 2, add x squared plus 6x plus 4, and we get 2x squared plus 6x plus 2 as our final answer.

Here is a question for you to try.

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Here are the answers.

For part a, fh of x is 8 over x squared, and hf of x is 64 over x squared.

And when you add these fractions together, you get 72 over x squared.

For part b, we have to apply three functions, h of x first, then g of x, and then f of x.

The order is really important.

Here is a final question for you to have a go at.

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Here is the answer.

What a lovely question to finish with.

We get x squared minus 6x plus 5 equals x squared plus 1, which rearranges to 6x equals 12, or you may have got minus 6x equals 12.

And then we get x equals two.

That's all for this lesson.

Remember to take the exit quiz before you leave.

Thank you for watching.