A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 2: Functions and graphs · Lesson 2

Functions, mappings and composite functions

119 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-functions-mappings-and-composite-functions/

Date

Do Now

Quick questions to start.

A1
Needed today
Given $f (x) = 3 x^{2} - 2$ , find $f (- 2)$ and $f (\frac{1}{3})$ .
[2 marks]
Answer
A2
Needed today
Write $x^{2} - 6 x + 11$ in the form $(x - p)^{2} + q$ . Hence state the least value of $x^{2} - 6 x + 11$ .
[3 marks]
Answer
A3
From chapter 1
Solve $2 x^{2} - 5 x - 3 = 0$ .
[2 marks]
Answer
A4
Needed today
Given $g (x) = 2 x + 1$ , write $g (x + 3)$ in its simplest form.
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

The function $f$ is defined by $f (x) = x^{2} - 4 x + 1$ , $x \in R$ . Find the range of $f$ .

Hint Complete the square. The least value of a squared bracket is

0

B1
The function $f$ is defined by $f (x) = x^{2} + 6 x + 4$ , $x \in R$ . Find the range of $f$ .
[3 marks]
Answer

Example 2

$f (x) = 2 x - 1$ and $g (x) = x^{2} + 3$ , $x \in R$ . Find

(a) $f g (2)$

(b) $g f (x)$ in its simplest form.

Hint In

f g (x)

the function written next to

x

is applied first:

f g (x) = f (g (x))

B2
$f (x) = 3 x + 2$ and $g (x) = x^{2} - 1$ , $x \in R$ . Find
(a) $f g (3)$
(b) $g f (x)$ in its simplest form.
[4 marks]
Answer

Example 3

$f (x) = 4 x - 3$ , $x \in R$ , and $g (x) = \frac{1}{x + 2}$ , $x > - 2$ . Solve $f g (x) = 5$ .

Hint Write

f g (x)

as one expression, then solve. Check that your answer is in the domain of

g

B3
$f (x) = 2 x + 5$ , $x \in R$ , and $g (x) = \frac{1}{x - 1}$ , $x > 1$ . Solve $f g (x) = 9$ .
[4 marks]
Answer

Example 4

$f (x) = x - 5$ , $x \in R$ , and $g (x) = x$ , $x \geq 0$ . Find the largest possible domain of $g f$ .

Hint

g

can only take inputs that are

0

or more. So the output of

f

must be

0

or more.

B4
$f (x) = 4 - x^{2}$ , $x \in R$ , and $g (x) = x$ , $x \geq 0$ .
(a) Explain why $g f (5)$ is not defined.
(b) Find the largest possible domain of $g f$ .
[4 marks]
Answer

Example 5

$f (x) = {x^{2}, 3 x, x < 0 x \geq 0$ . Sketch $y = f (x)$ and solve $f (x) = 6$ .

Hint Solve in each piece, then keep only the answers that lie in that piece's interval.

B5
$f (x) = {2 x + 1, 5 - 2 x, x \leq 1 x > 1$ . Solve $f (x) = 2$ .
[4 marks]
Answer

Practice

A-Level

C1
For each mapping, state whether it is a function. If it is, state whether it is one-to-one or many-to-one. Give a reason each time.
(a) $y = \frac{1}{x - 2}$ , $x \in R$
(b) $y = x^{2}$ , $x \in R$
(c) $y^{2} = x$ , $x \geq 0$
[5 marks]

C2
$f (x) = x + 4$ and $g (x) = 3 x^{2}$ , $x \in R$ . Lily works out $f g (1)$ . She writes: " $f (1) = 5$ , then $g (5) = 75$ , so $f g (1) = 75$ ." Explain her mistake and find the correct value of $f g (1)$ .
[4 marks]
Answer
C3
$f (x) = 2 x - 3$ and $g (x) = x^{2}$ , $x \in R$ . Solve $f g (x) = g f (x)$ , giving your answers in surd form.
[5 marks]
Answer
C4
$f (x) = x^{2} + 2$ , $x \in R$ , and $g (x) = \frac{6}{x}$ , $x > 0$ . Find $g f (x)$ and the range of $g f$ .
[4 marks]
Answer
C5
$h (x) = x^{2} - 4 x + 7$ , $0 \leq x \leq 5$ . Ben says the range of $h$ is $7 \leq h (x) \leq 12$ , because $h (0) = 7$ and $h (5) = 12$ . Explain why Ben is wrong and find the correct range.
[5 marks]
Answer
C6
$f (x) = \frac{3}{x - 1}$ , $x \in R$ , $x \neq = 1$ .
(a) Show that $f f (x) = \frac{3 ( x - 1 )}{4 - x}$ .
(b) Hence solve $f f (x) = x$ , giving your answers in surd form.
[6 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The functions $f$ and $g$ are defined by $f (x) = x^{2} - 6 x + 5$ , $x \in R$ , and $g (x) = 2 x + 1$ , $x \in R$ .
(a)
Find the range of $f$ .
[3 marks]
Answer
(b)
Show that $f g (x) = 4 x^{2} - 8 x$ .
[2 marks]
(c)
Solve $f g (x) = 12$ .
[3 marks]
Answer
(d)
Find the range of $f g$ .
[2 marks]
Answer
Total for D1: 10 marks

D2
$f (x) = x^{2} + 1$ , $x \in R$ , and $g (x) = x - 3$ , $x \geq 3$ . Ana says: " $g$ has domain $x \geq 3$ , so the domain of $g f$ is also $x \geq 3$ ."
Ana is wrong.
(a)
Explain the mistake Ana makes.
[2 marks]
(b)
Find the largest possible domain of $g f$ .
[4 marks]
Answer
(c)
Solve $g f (x) = 6$ .
[3 marks]
Answer
Total for D2: 9 marks
D3
The function $f$ is defined by $f (x) = x^{2} - 2 x - 3$ , $x \in R$ . Find, in exact form, every solution of the equation $f f (x) = 3$ .
[7 marks]
Answer
D4
The function $f$ is defined by $f (x) = {x + 4, x^{2} + 2, x < - 1 x \geq - 1$
(a)
Find the value of $f (- 3) + f (2)$ .
[2 marks]
Answer
(b)
Solve $f (x) = 11$ .
[3 marks]
Answer
(c)
Find $f f (- 2)$ .
[2 marks]
Answer
(d)
Find the set of values of $k$ for which the equation $f (x) = k$ has exactly three solutions.
[4 marks]
Answer
Total for D4: 11 marks
D5
$f (x) = 2 x + 3$ , $x \in R$ , and $g (x) = \frac{12}{x + 1}$ , $x > - 1$ .
(a)
Find $g f (x)$ in its simplest form, and state its domain.
[3 marks]
Answer
(b)
Solve $g f (x) = f (x)$ .
[4 marks]
Answer
(c)
State the range of $g f$ .
[2 marks]
Answer
Total for D5: 9 marks
D6
$f (x) = 2 x^{2} - 8 x + 5$ , $- 1 \leq x \leq 5$ .
(a)
Find the range of $f$ .
[4 marks]
Answer
(b)
Find the set of values of $k$ for which $f (x) = k$ has two distinct solutions.
[3 marks]
Answer
Total for D6: 7 marks

Extension

Stretch

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

E1
The function $f$ is defined by $f (x) = x^{2} - 4 x$ , $x \in R$ . Find the set of values of $k$ for which the equation $f f (x) = k$ has exactly four real solutions.
[9 marks]
Hint 1 · What to try
Let $y = f (x)$ . First solve $f (y) = k$ for $y$ , then ask how many $x$ give each value of $y$ .
Hint 2 · The first line
$f (y) = k$ gives $y = 2 \pm k + 4$ . The range of $f$ is $f (x) \geq - 4$ , and $f (x) = y$ has two solutions only when $y > - 4$ .
Hint 3 · The full method
$y = 2 + k + 4$ is always more than $- 4$ . Find when $y = 2 - k + 4$ is more than $- 4$ , and when the two values of $y$ are different.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can say whether a mapping is a function, and whether it is one-to-one or many-to-one.
I can find the range of a function on a given domain, using its turning point.
I can work out fg(x) and gf(x), applying g first in fg(x).
I can find the domain and range of a composite function and solve equations involving it.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers