 # Use Composite Functions

In this worksheet, students will find outputs from composite functions involving up to three separate functions carried out in a set sequence. 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:   ### QUESTION 1 of 10

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

As an example, let's say that f(x) = x + 1 and g(x) = x² - 3.  Find fg(5).

fg(x) means carry out g(x) first and then we need to put our output into f(x).

The first function we need to deal with is the one closest to (x), and work through them from left to right.

First, let's find g(5):

g(5) = 5² - 3 = 25 - 3 = 22

Now let's put our output into f(x):

f(22) = 22 + 1 = 23

So fg(5) = 23

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

Just to prove this, let's consider gf(5).

f(5) = 5 + 1 = 6

g(6) = 6² - 3 = 36 - 3 = 33

So gf(5) = 33

In this activity, we will find outputs from composite functions involving up to three separate functions carried out in a sequence, correctly prioritising their order based on their proximity to (x).

Consider the functions below:

f(x) = x + 3

g(x) = 2x

Find gf(5).

16

13

10

8

Consider the functions below:

f(x) = 2x + 4

g(x) = 11 - x

Match each composite function to their correct outputs.

## Column B

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

Consider the functions below:

f(x) = 3x - 1

g(x) = x2

What is the value of fg(4)?

Consider the functions below:

f(x) = 3x²

g(x) = x - 1

h(x) = 3x + 1

Match each composite function to their correct outputs.

## Column B

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

Consider the functions below:

f(x) = 10 - x

g(x) = x²

h(x) = 12 - 2x

Tick all the composite functions in the list below which will give an output of 2.

hf(5)

fg(3)

gh(4)

fh(0)

Consider the functions below:

f(x) = 5x

g(x) = 3x - 2

Using these functions, what is the correct expression for fg(x)?

15x - 10

15x2 - 10x

15x2 - 2

15x - 2

Consider the functions below:

f(x) = x²

g(x) = 2x - 5

Using these functions, what is the correct expression for gf(2x)?

4x² - 5

16x² - 30x + 25

8x² - 5

16x² - 25

Consider the functions below:

f(x) = 2x - 3

g(x) = x³

h(x) = 7 - 2x

Match each composite function to their correct expression.

## Column B

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

Consider the functions below:

f(x) = 4x

g(x) = 2x²

h(x) = x - 3

Using these functions, what is the correct expression for fhg(2x)?

16x² - 12

32x² - 12

16x² - 3

32x² - 3

Consider the functions below:

f(x) = x²

g(x) = 5 - x

h(x) = 3 - 4x

Using these functions, what is the correct expression for hgf(x²)?

4x4 - 15

4x4 + 15

4x2 - 15

15 - 4x4

• Question 1

Consider the functions below:

f(x) = x + 3

g(x) = 2x

Find gf(5).

16
EDDIE SAYS
Which function do we need to tackle first here? That's right, it's f(5) as this is the closest function to the (5). f(5) = 5 + 3 = 8 Now let's substitute 8 for x in g(x): g(8) = 2 × 8 = 16 So gf(5) = 16 How was that? Review the Introduction before moving on if you found those steps tricky to recall.
• Question 2

Consider the functions below:

f(x) = 2x + 4

g(x) = 11 - x

Match each composite function to their correct outputs.

## 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 brackets first. gg(2) looks a little different, but you just need to follow the same method. Find g(2) first: 11 - 2 = 9 Now put 9 back into g again: 11 - 9 = 2 So gg(2) = 2 Can you use this method and example to match the remaining three pairs independently?
• Question 3

Consider the functions below:

f(x) = 3x - 1

g(x) = x2

What is the value of fg(4)?

47
EDDIE SAYS
g is closest to (4) so we need to calculate this first. g(4) = 4² = 4× 4 = 16 (Remember that squaring is the same as multiplying a number by itself.) Now let's put this output into f(x): f(16) = 3 × 16 - 1 = 48 - 1 = 47 So fg(4) = 47 Did you type that number in correctly?
• Question 4

Consider the functions below:

f(x) = 3x²

g(x) = x - 1

h(x) = 3x + 1

Match each composite function to their correct outputs.

## 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 closest to the brackets first? Make sure you also follow the order of operations (BIDMAS or BODMAS), otherwise you will not reach a correct answer. Let's look at one example together: fh(2). Find h(2) first: 3 × 2 + 1 = 6 + 1 = 7 Now substitute 7 into f(x): f(7) = 3 × 7² = 3 × 49 = 147 The rules of BIDMAS mean that we need to work out the square here (I for 'Indices') before the multiplication (M). Can you use this example and the rules of BIDMAS to help you match the other functions and outputs independently?
• Question 5

Consider the functions below:

f(x) = 10 - x

g(x) = x²

h(x) = 12 - 2x

Tick all the composite functions in the list below which will give an output of 2.

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 So hf(5) = 2 fh(0) gives an answer of -2. fg(3) = 1, whereas gh(4) = 16. Did you find the only correct answer from the options?
• Question 6

Consider the functions below:

f(x) = 5x

g(x) = 3x - 2

Using these functions, what is the correct expression for fg(x)?

15x - 2
EDDIE SAYS
In this question, no number has been provided to work with so we need to use algebra instead. In this question, we have been asked to use 'x' instead of a number. First let's find f(x). This is just 5x. Now let's put '5x' into g(x): 3(5x) - 2 = 15x - 2 So fg(x) = 15x - 2
• Question 7

Consider the functions below:

f(x) = x²

g(x) = 2x - 5

Using these functions, what is the correct expression for gf(2x)?

8x² - 5
EDDIE SAYS
On this occasion, we need to substitute '2x' for 'x' in the functions, starting with f(x). f(2x) = (2x)² = 4x² Now let's use 4x² in the place of x in g(x): g(4x²)= 2(4x²) - 5 And then let's simplify: 2(4x²) - 5 = 8x² - 5
• Question 8

Consider the functions below:

f(x) = 2x - 3

g(x) = x³

h(x) = 7 - 2x

Match each composite function to their correct expression.

## Column B

fg(x)
2x³ - 3
hg(x)
7 - 6x³
fh(2x)
11 - 8x
hf(x²)
13 - 4x²
EDDIE SAYS
We need to follow the same process from the last question, and substitute the different algebraic values in to each of the relevant functions. Let's have a look at two hardest questions together. 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 - 4x² + 6 = 13 - 4x² Can you find the remaining two matches? Remember to always start with the function closest to the brackets and then substitute the output from this function into the other.
• Question 9

Consider the functions below:

f(x) = 4x

g(x) = 2x²

h(x) = x - 3

Using these functions, what is the correct expression for fhg(2x)?

32x² - 12
EDDIE SAYS
This question looks difficult, but it's hardly different than the previous ones so don't worry! Here we need to carry out three functions one after another. Let's do this slowly to make sure we 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 How was that?
• Question 10

Consider the functions below:

f(x) = x²

g(x) = 5 - x

h(x) = 3 - 4x

Using these functions, what is the correct expression for hgf(x²)?

4x4 - 15
EDDIE SAYS
Another tricky question! Did you break it down into steps? f(x²)= (x²)² = x4 g(x4) = 5 - x4 h(5 - x4) = 5 - 4(5 - x4) = 5 - 20 + x4 = - 15 + x4 = x4 - 15 Great work completing this activity! You may want to revise BIDMAS (or BODMAS) if you found that you were rusty on these skills at all.
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