GCSE Maths · Higher · Chapter 24: Algebraic fractions and functions · Lesson 3

Functions and composite functions

Grades 6–9About 50 minutesCalculator allowed39 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Work out $4 x^{2} + 6$ when $x = - 3$ .
[1 mark]
Answer
A2
From chapter 22
Write down the maximum value of $y = 5 sin x$ .
[1 mark]
Answer
A3
From chapter 8
Expand $(x - 6) (x + 5)$ .
[1 mark]
Answer
A4
From chapter 9
A circle has radius 12 cm. Find its circumference in terms of $π$ .
[1 mark]
Answer

Grades 6–8

Your teacher works through each example with you. Then try the one beside it.

Example 1

$f (x) = 3 x - 2$ and $g (x) = x^{2} + 1$ . Work out $f g (2)$ and $g f (2)$ , then write $f g (x)$ in terms of $x$ .

Hint

f g (2)

means do

g

first, then

f

. The function nearest the bracket goes first.

B1
$f (x) = 2 x + 5$ and $g (x) = x^{2} - 6$ . Work out $g f (- 1)$ , then write $f g (x)$ in terms of $x$ .
[3 marks]
Answer $g f (- 1) = \dots\dots$ $f g (x) = \dots\dots\dots\dots$

Example 2

$f (x) = \frac{2 x + 9}{5}$ . Find $f^{- 1} (x)$ .

Hint Write

y = f (x)

, make

x

the subject, then swap

x

and

y

back.

B2
$g (x) = 7 - 2 x$ . Find $g^{- 1} (x)$ , then work out $g^{- 1} (- 5)$ .
[3 marks]
Answer $g^{- 1} (x) = \dots\dots\dots$ $g^{- 1} (- 5) = \dots\dots$

Grades 6–8

C1
$h (x) = x^{2} + 4 x$ . Solve $h (x) = 21$ .
[2 marks]
Answer
C2
$f (x) = x^{2} - 3 x$ . Write $f (x + 2)$ in its simplest form.
[2 marks]
Answer

C3
$f (x) = 4 x - 3$ . Find $f f (x)$ in its simplest form.
[2 marks]
Answer
C4
$f (x) = \frac{5}{x - 1} + 2$ . Find $f^{- 1} (x)$ .
[3 marks]
Answer
C5
$f (x) = 2 x - 1$ and $g (x) = x^{2} + 3$ . Solve $f g (x) = 37$ .
[3 marks]
Answer
C6
$f (x) = x^{3} - 5$ . Work out $f^{- 1} (59)$ .
[2 marks]
Answer

Grades 7–9

D1
$f (x) = 2 x + 6$ . Lena says, " $f^{- 1} (4) = \frac{1}{f ( 4 )} = \frac{1}{14}$ ."
Explain why Lena is wrong, and work out the correct value of $f^{- 1} (4)$ .
[2 marks]

D2
A bank changes pounds into euros. It uses $e (x) = 1.15 x$ to change $x$ pounds into euros. It then takes a fee of 4 euros, using $h (x) = x - 4$ .
(a)
Work out $h e (200)$ , the number of euros paid out for £200.
[1 mark]
Answer€ $\dots\dots$
(b)
Find $(h e)^{- 1} (x)$ .
[2 marks]
Answer
(c)
Kim is paid €341. Work out how many pounds she changed.
[2 marks]
Answer£ $\dots\dots$
Total for D2: 5 marks
D3
$f (x) = a x + b$ , where $a$ and $b$ are numbers. $f (2) = 11$ and $f^{- 1} (26) = 7$ . Work out the values of $a$ and $b$ .
[4 marks]
Answer $a = \dots\dots$ $b = \dots\dots$

Grade 9 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use function notation and solve $f (x) = k$ .
I can work out composite functions like $f g (x)$ , in the right order.
I can find and use an inverse function.

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