AQA Level 2 Further Maths · Level 2 Certificate · Chapter 3: Algebra III · Lesson 2

Composite functions

About 80 minutesNo calculator70 marks

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The first two questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
$g (x) = \frac{12}{x + 1}$ . Solve $g (x) = 4$ .
[2 marks]
Answer $x = \dots\dots$
A2
Needed today
Expand and simplify $(2 x - 3)^{2} + 4$ .
[2 marks]
Answer
A3
Last lesson
$f (x) = 7 - 3 x$ with domain $- 2 \leq x \leq 4$ . Write down the range of $f (x)$ .
[2 marks]
Answer $\dots\dots \leq f (x) \leq \dots\dots$
A4
From chapter 2
Write $x^{2} - 10 x + 4$ in the form $(x - a)^{2} - b$ .
[2 marks]
Answer $(x - \dots\dots)^{2} - \dots\dots$

Example, then your turn

Level 2

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

Example 1

$f (x) = 3 x - 1$ and $g (x) = x^{2} + 2$ . Write $f g (x)$ and $g f (x)$ in their simplest forms.

Hint

f g (x)

means

f (g (x))

: do

g

first, then put the result into

f

B1
$f (x) = 2 x + 5$ and $g (x) = x^{2} - 3$ . Write $f g (x)$ and $g f (x)$ in their simplest forms.
[4 marks]
Answer $f g (x) = \dots\dots g f (x) = \dots\dots$

Example 2

$f (x) = 3 x - 2$ , $g (x) = \frac{1}{x}$ and $h (x) = x + 3$ . Write $f g h (x)$ and $f f (x)$ in their simplest forms.

Hint In

f g h (x)

the function next to

x

works first:

h

, then

g

, then

f

f f (x)

means

f (f (x))

B2
$f (x) = 2 x - 1$ , $g (x) = \frac{1}{x}$ and $h (x) = x^{2}$ . Write $g f h (x)$ and $f f (x)$ in their simplest forms.
[4 marks]
Answer $g f h (x) = \dots\dots f f (x) = \dots\dots$

Example 3

$h (x) = \frac{5}{x ^{2} + 1}$ . Write $h (x)$ in the form $f g (x)$ , stating $f (x)$ and $g (x)$ .

Hint Ask what is done to

x

first. That step is

g

; what is done to the result is

f

B3
$k (x) = 2 (x - 3)^{4}$ . Write $k (x)$ in the form $pq r (x)$ , stating $p (x)$ , $q (x)$ and $r (x)$ .
[3 marks]
Answer $p (x) = \dots\dots q (x) = \dots\dots r (x) = \dots\dots$

Example 4

$f (x) = x^{2} - 4$ and $g (x) = 2 x + 1$ . Solve $f g (x) = 21$ .

Hint Write

f g (x)

first. A square equal to

25

gives two solutions.

B4
$f (x) = x^{2} + 3$ and $g (x) = 3 x - 2$ . Solve $f g (x) = 52$ .
[3 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$

Example 5

$f (x) = x - 2$ and $g f (x) = x^{2} - 4 x - 1$ . Work out $g (x)$ .

Hint Write

g f (x)

in terms of

(x - 2)

, the output of

f

. Completing the square does it.

B5
$f (x) = x + 5$ and $g f (x) = x^{2} + 10 x + 30$ . Work out $g (x)$ .
[3 marks]
Answer $g (x) = \dots\dots$

Practice

Level 2

C1
$f (x) = x^{2} - 2 x$ and $g (x) = 5 - x$ . Work out $f g (- 1)$ and $g f (- 1)$ .
[3 marks]
Answer $f g (- 1) = \dots\dots g f (- 1) = \dots\dots$
C2
$f (x) = x - 4$ and $g (x) = 2 x$ . Write each of these in its simplest form: $f (x) \times g (x)$ , then $f g (x)$ .
[2 marks]
Answer $f (x) \times g (x) = \dots\dots f g (x) = \dots\dots$

C3
$f (x) = x$ for $x \geq 0$ , and $g (x) = 10 - 2 x$ . Work out the domain of $f g (x)$ .
[2 marks]
Answer $\dots\dots\dots\dots$
C4
$f (x) = x^{2}$ for all $x$ , and $g (x) = 3 - x$ with domain $1 \leq x \leq 5$ . Work out the range of $f g (x)$ .
[3 marks]
Answer $\dots\dots \leq f g (x) \leq \dots\dots$
C5
$f (x) = 2 x + a$ and $g (x) = x^{2} - 1$ , where $a$ is a constant. Given that $f g (3) = 11$ , find $a$ . Then work out $g f (3)$ .
[3 marks]
Answer $a = \dots\dots g f (3) = \dots\dots$

Exam-style questions

AQA exam style

D1
$f (x) = x$ for $x \geq 0$ . $g$ is a function such that $g f (x) = 2 x + 3 x$ . Work out an expression for $g (x)$ .
[2 marks]
Answer $g (x) = \dots\dots$

D2
$h (x) = 2 x^{2} - 7$ . $k (x) = c x$ , where $c$ is a positive constant, and $hk (x) = 18 x^{2} - 7$ . Work out an expression for $k h (x)$ .
[3 marks]
Answer $k h (x) = \dots\dots$
D3
$f (x) = 2 x^{2} - 1$ for all $x$ . $g (x) = x + 3$ for all $x$ .
(a)
Work out the value of $g f (2)$ .
[2 marks]
Answer $\dots\dots$
(b)
Show that $f g (x) = 2 x^{2} + 12 x + 17$ .
[2 marks]
(c)
Hence solve $f g (x) = 13$ . Give your answers in the form $p \pm q$ , where $p$ and $q$ are integers.
[3 marks]
Answer $x = \dots\dots \pm \dots\dots$
Total for D3: 7 marks
D4
$f (x) = x^{2}$ and $g (x) = x - 6$ . Mia says, " $f g (x) = f (x) \times g (x)$ , so $f g (x) = x^{3} - 6 x^{2}$ ."
Mia is wrong.
(a)
Explain Mia's mistake, and write $f g (x)$ correctly.
[2 marks]
Answer $f g (x) = \dots\dots$
(b)
Solve $f g (x) = g f (x)$ .
[3 marks]
Answer $x = \dots\dots$
Total for D4: 5 marks
D5
$f (x) = 3 x + p$ and $g (x) = x^{2} + q$ , where $p$ and $q$ are integers. $f g (x) = 3 x^{2} + 7$ and $g f (1) = 4$ . Work out the values of $p$ and $q$ . You must show your working.
[5 marks]
Answer $p = \dots\dots q = \dots\dots$
D6
$f (x) = \frac{1}{x - 2}$ for $x \neq = 2$ . $g (x) = 3 x + 4$ for all $x$ .
(a)
Which value of $x$ must be excluded from the domain of $f g (x)$ ? Circle your answer. $- 2 - \frac{2}{3} \frac{2}{3} 2$
[1 mark]
(b)
Solve $g f (x) = 10$ .
[2 marks]
Answer $x = \dots\dots$
(c)
Show that $f f (x) = \frac{x - 2}{5 - 2 x}$ .
[3 marks]
Total for D6: 6 marks

Extension

Stretch

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

E1
$g (x) = a x + b$ , where $a$ and $b$ are constants, and $g g (x) = 9 x - 16$ . Find every possible function $g (x)$ .
[4 marks]
Hint 1 · What to try
Work out $g g (x)$ in terms of $a$ and $b$ , then match it with $9 x - 16$ .
Hint 2 · The first line
$g g (x) = a (a x + b) + b = a^{2} x + ab + b$ , so $a^{2} = 9$ and $ab + b = - 16$ .
Hint 3 · The full method
$a = 3$ gives $4 b = - 16$ , so $b = - 4$ . $a = - 3$ gives $- 2 b = - 16$ , so $b = 8$ . Both work.
Answer $g (x) = \dots\dots$ or $g (x) = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write $f g (x)$ , $g f (x)$ , $f f (x)$ and $f g h (x)$ as expressions, and know the order matters.
I can split a function into a composite of simpler functions.
I can solve equations involving composite functions and find unknown constants.
I can find a function from a composite, and the domain and range of a composite.

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