The smart way to improve grades

Comprehensive & curriculum aligned

Affordable pricing from £10/month

Use Composite Functions

In this worksheet, students will learn about using composite functions.

'Use Composite Functions' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Notation, Vocabulary and Manipulation, Language of Functions

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Composite functions are 2 (or more) functions carried out in sequence.

Let's say that f(x) = x + 1 and g(x) = x² - 3

fg(x) means carry out g(x) first and then put your output into f(x). The first function you need to deal with is th one that is next to (x) and work through them from left to right.

fg(5) would be:

First find g(5).

g(5) = 5² - 3 = 25 - 3 = 22

Now put your output into f(x)

f(22) = 22 + 1 = 23

So fg(5) = 23.

If we change the order, the answer will be different, so always check carefully which function you need to start with.

gf(5) would go like this

f(5) = 5 + 1 = 6

g(6) = 6² - 3 = 36 - 3 = 33

gf(5) = 33

f(x) = x + 3

g(x) = 2x

Find gf(5).

16

13

10

8

f(x) = 2x + 4

g(x) = 11 - x

Match the composite functions to their correct values.

Column A

Column B

fg(3)
20
gf(3)
22
fg(2)
2
gg(2)
1

f(x) = 3x - 1

g(x) = x2.

What is the value of fg(4)?

121

47

11

23

f(x) = 3x²

g(x) = x - 1

h(x) = 3x + 1

Match the composite functions to the appropriate values.

Column A

Column B

gf(2)
6
gh(2)
11
hg(2)
147
fh(2)
4

f(x) = 10 - x

g(x) = x²

h(x) = 12 - 2x

Tick all the composite functions which will give you an answer of 2.

hf(5)

fg(3)

gh(4)

fh(0)

What is the correct expression for fg(x) if f(x) = 5x and g(x) = 3x - 2?

15x - 10

15x² - 10x

15x² - 2

15x - 2

f(x) = x²

g(x) = 2x - 5

Find the expression for gf(2x).

4x² - 5

16x² - 30x + 25

8x² - 5

16x² - 25

f(x) = 2x - 3

g(x) = x³

h(x) = 7 - 2x

Match the expressions to the correct composite functions.

Column A

Column B

fg(x)
11 - 8x
hg(x)
2x³ - 3
fh(2x)
13 - 4x²
hf(x²)
7 - 6x³

f(x) = 4x

g(x) = 2x²

h(x) = x - 3

Find fhg(2x).

16x² - 12

32x² - 12

16x² - 3

32x² - 3

f(x) = x²

g(x) = 5 - x

h(x) = 3 - 4x

Find the value of hgf(x²).

4x&sup4; - 15

4x&sup4; + 15

4x² - 15

15 - 4x⊃

  • Question 1

f(x) = x + 3

g(x) = 2x

Find gf(5).

CORRECT ANSWER
16
EDDIE SAYS
To find the value of gf(5), first work out f(5). f(5) = 5 + 3 = 8 Now put 8 into g(x). g(8) = 2 × 8 = 16. So gf(5) = 16.
  • Question 2

f(x) = 2x + 4

g(x) = 11 - x

Match the composite functions to their correct values.

CORRECT ANSWER

Column A

Column B

fg(3)
20
gf(3)
1
fg(2)
22
gg(2)
2
EDDIE SAYS
Remember to work out the function next to the bracket first! gg(2) looks a little different, but you just need to follow the method. Find g(2) first. 11 - 2 = 9 Now put 9 into g. 11 - 9 = 2. So gg(2) = 2.
  • Question 3

f(x) = 3x - 1

g(x) = x2.

What is the value of fg(4)?

CORRECT ANSWER
47
EDDIE SAYS
g(4) = 4² = 4× 4 = 16 (remember that squaring is timesing my itself!) f(16) = 3 × 16 - 1 = 48 - 1 = 47 fg(4) = 47
  • Question 4

f(x) = 3x²

g(x) = x - 1

h(x) = 3x + 1

Match the composite functions to the appropriate values.

CORRECT ANSWER

Column A

Column B

gf(2)
11
gh(2)
6
hg(2)
4
fh(2)
147
EDDIE SAYS
Did you remember to first work out the function that is right next to the number? Make sure you follow the order of operations, otherwise you will not get a correct answer. In fh(2), find h(2) first: 3 × 2 + 1 = 6 + 1 = 7 Now work out f(7) = 3 × 7² = 3 × 49 = 147.
  • Question 5

f(x) = 10 - x

g(x) = x²

h(x) = 12 - 2x

Tick all the composite functions which will give you an answer of 2.

CORRECT ANSWER
hf(5)
EDDIE SAYS
Only hf(5) gives an answer of 2. f(5) = 10 - 5 = 5 h(5) = 12 - 2 × 5 = 12 - 10 = 2 fh(0) gives an answer of -2, so it is incorrect. fg(3) = 1, whereas gh(4) = 16, so these are also incorrect.
  • Question 6

What is the correct expression for fg(x) if f(x) = 5x and g(x) = 3x - 2?

CORRECT ANSWER
15x - 2
EDDIE SAYS
There is no number to put into the functions, so you just need to work with algebra. First find f(x). This is just 5x. Now put this into g(x). 3(5x) - 2 = 15x - 2 So fg(x) = 15x - 2.
  • Question 7

f(x) = x²

g(x) = 2x - 5

Find the expression for gf(2x).

CORRECT ANSWER
8x² - 5
EDDIE SAYS
Did you remember to square the 2 in front of the x? f(2) = (2x)² = 4x² g(4x²)= 2(4x²) - 5 = 8x² - 5.
  • Question 8

f(x) = 2x - 3

g(x) = x³

h(x) = 7 - 2x

Match the expressions to the correct composite functions.

CORRECT ANSWER

Column A

Column B

fg(x)
2x³ - 3
hg(x)
7 - 6x³
fh(2x)
11 - 8x
hf(x²)
13 - 4x²
EDDIE SAYS
Let's have a look at two hardest questions. fh(2x) h(2x) = 7 - 2(2x) = 7 - 4x f(7 - 4x)= 2(7 - 4x) - 3 = 14 - 8x - 3 = 11 - 8x hf(x²) f(x²) = 2(x²) - 3 = 2x² - 3 h(2x³ - 3) = 7 - 2(2x² - 3) = 7 - 4x&up2; + 6 = 13 - 4x²
  • Question 9

f(x) = 4x

g(x) = 2x²

h(x) = x - 3

Find fhg(2x).

CORRECT ANSWER
32x² - 12
EDDIE SAYS
Don't worry! This question looks difficult, but it's hardly different than the previous ones. Here you just need to carry out three functions one after another. Do it slowly to make sure you don't make any mistakes. g(2x) = 2(2x)² = 2 × 4x² = 8x² h(8x²) = 8x² - 3 f(8x² - 3) = 4(8x² - 3) = 32x² - 12
  • Question 10

f(x) = x²

g(x) = 5 - x

h(x) = 3 - 4x

Find the value of hgf(x²).

CORRECT ANSWER
4x&sup4; - 15
EDDIE SAYS
Another tricky question! Did you break it down into steps? f(x²)= (x²)² = x&sup4; g(x&sup4;) = 5 - x&sup4; h(5 - x&sup4;) = 5 - 4(5 - x&sup4;) = 5 - 20 + x&sup4; = - 15 + x&sup4; = x&sup4; - 15
---- OR ----

Sign up for a £1 trial so you can track and measure your child's progress on this activity.

What is EdPlace?

We're your National Curriculum aligned online education content provider helping each child succeed in English, maths and science from year 1 to GCSE. With an EdPlace account you’ll be able to track and measure progress, helping each child achieve their best. We build confidence and attainment by personalising each child’s learning at a level that suits them.

Get started
laptop

Start your £1 trial today.
Subscribe from £10/month.