﻿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 and a function g and what part A is asking us to do is take the function g , substitute negative four into it, and then whatever answer we get, we're going to put into the function f . So the order is very important here. So we're going to be doing the g function first, because that's the one that's immediately next to the bracket. And then the function f is going to act on that answer. Okay, so let's start by substituting negative four into our function g . So we get two times negative four, add five, which gives us negative three. But we haven't finished yet, because we need to put that negative three into the function of x now, so we get negative three, multiply by seven, and that is negative 21. So there we've got the function g , and then the function f 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 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 seven times two, 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 seven times x or seven times 14, And we get our answer of 98. Here are some questions for you to try. Pause the video to complete the task and restart when you are finished. Here are the answers. Be very careful about the order in which you apply the functions. Start with the function closest to the brackets 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 . So again, we want to put four into our 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 . So let's put four into f , and we get four plus three equals seven and then we put seven into g . So we get seven squared minus five equals 42. And that's our answer to that composite function. Next, we're going to look at applying g twice to x. So we want to substitute x into g . 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 again, so this time we've got g squared minus five. So we're going to get x minus five all squared, and then takeaway 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. Don't forget to take away that five on the end. It's really important. It's a common mistake that people will forget about the taking away that five at the end, if you've got an extra term at the end of the function. So we simplify that and we get x squared minus 10 x plus 20. Here are some questions for you to try. Pause the video to complete the task and restart when you finished. Here are the answers. The last three questions are a little bit trickier, as you substituted in more than one term into the second function. I find it useful to put brackets around the terms I'm 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 seven all squared, which gives x the power of four, take away 14x squared plus 49 and then take away the seven 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 that is then being acted upon by g and finally, f , so fgh is what we said. So the first thing we do is we apply h to x, so substitute x into h . And that's quite straightforward because we just substitute in x in, so we get three divided by x. We now want to take this answer and apply g to it. So we've got g of three divided by x, and that would give us three over x all squared takeaway five. So if we simplify that, we get nine over x squared, takeaway five. And finally, we want to apply f to this answer, so we substitute in nine over x squared take with 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, takeaway two. For part B, we want fg added to gf . And be very careful here because these don't give the same answers and you'll see this shortly. So we start with the first part of the addition fg , and we apply the function g . 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 , so that we can add it on to this first function of a function that we found. So we look for f first, so we substitute f into there and we get x plus three, and then we apply g . So we put x plus three into the function g , and we get x plus three, all squared, take away five. And we expand that to get x squared plus six x plus four, and then we need to add together, these two answers. Now notice we've got different answers, because we were applying the functions in different artists. There are some cases where we do get the same answer, but not always, 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 two, add x squared plus six x plus four, and we get two x squared plus six x plus two as our final answer. Here's a question for you to try. Pause the video to complete the task and restart when you're finished. Here are the answers. For part A fh is eight over x squared, and hf is 65 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 first, then g and then f . The order is really important. Here is a final question for you to have a go at. Pause the video now to complete the task and restart when you are finished. Here is the answer. What a lovely question to finish with, we get x squared minus six x plus five equals x squared plus one, which rearranges to six x equals 12, or you may have gotten minus six x 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.

In this lesson, we will apply more than one function to a number or a variable using substitution, knowledge of expanding brackets and also apply these skills to solving equations such as fg(x) = h(x)

Maths

Y11 Maths (Core)

Key Stage 4

Year 11

Composite functions

Y11 Maths (Higher)

Functions

Algebra (A)

THEME

UNIT

Find a particular value of f(x)

Solve equations using f(x) =

Find inverse functions

Express Maths

Y11 Express Maths (Higher)

Algebraic Proof & Functions

Prove by counter example

Prove an expression will be a multiple of a given number

Consecutive number proofs

Odd and even number proofs

Lesson overview: Composite functions

Video

Presentation

Worksheet

Exit Quiz

Transcript

4 lessons in Functions:

8 lessons in Algebraic Proof & Functions: