iGCSE (2021 Edition)

# 4.02 Composite functions

Worksheet
Composite functions
1

Evaluate the following, given the functions:

\begin{aligned} f \left( x \right) &= 4 x^{3} \\ g \left( x \right) &= x + 5 \end{aligned}
a
f \left( - 2 \right)
b
g \left( f \left( - 2 \right) \right)
c
f \left( g \left( 0 \right) \right)
d
f \left( f \left( 0 \right) \right)
2

Evaluate f \left( g \left( 2 \right) \right), given the functions:

\begin{aligned} f \left( x \right) &= 4 x - 10 \\ g \left( x \right) &= 3 + \dfrac{3}{x}\end{aligned}
3

Consider the following functions:

\begin{aligned} f \left( x \right) = - 2 x - 6 \\ g \left( x \right) = 5 x - 7 \end{aligned}
a

Find f \left( 7 \right).

b

Hence, evaluate g \left( f \left( 7 \right) \right).

c

Find g \left( 7 \right).

d

Hence, evaluate f \left( g \left( 7 \right) \right).

e

Does f \left( g \left(x\right)\right) = g \left( f \left(x\right)\right) for all x?

4

Find the composite function f \left( g \left( x \right) \right) for the following:

a

f \left( x \right) = x^{3} \\ g \left( x \right) = 9 x - 4

b

f \left( x \right) = x^{2} \\ g \left( x \right) = x^{2} + 1

c

f \left( x \right) = \sqrt{x} \\ g \left( x \right) = 4 x - 3

d

f \left( x \right) = 4 x + 9 \\ g \left( x \right) = x^{3}

e

f \left( x \right) = 4 x - 3 \\ g \left( x \right) = x^{2}

5

Find the composite function f ^2\left( x \right) for the following:

a

f \left( x \right) = x^{2}

b

f \left( x \right) = x^{3}

c

f \left( x \right) = x+7

d

f \left( x \right) = 8x

e

f \left( x \right) = 3x-4

f

f \left( x \right) = \dfrac{x}{2}

g

f \left( x \right) = 5 x + 9

h

f \left( x \right) = 6 x +1

6

Evaluate the following, given the functions:

\begin{aligned} f \left(x\right) &= - 2 x + 2 \\ g \left(x\right) &= 4 x^{2} - 8 \\ r \left(x\right) &= - 3 x - 8 \end{aligned}
a

g \left( 6 \right)

b

f \left( g \left( 6 \right) \right)

c

r\left(f\left(g\left(6\right)\right)\right)

d

g\left(g\left(6\right)\right)

7

Consider the following functions:

\begin{aligned} f \left( x \right) &= - 2 x - 8 \\ g \left( x \right) &= 4 x^{2} - 4 \end{aligned}
a

Evaluate g \left( f \left( 6 \right) \right).

b

If h \left( x \right) is defined as f \left( g \left( x \right) \right), state the equation for h \left( x \right).

c

What type of functions are f \left( g \left( x \right) \right) and g \left( f \left( x \right) \right)?

8

Consider the following functions:

\begin{aligned} f \left(x\right) &= - 2 x + 6 \\ g \left(x\right) &= 3 x + 1 \end{aligned}
a

If r \left(x\right) is defined as f \left(x^{2}\right), state the equation for r \left(x\right).

b

Hence, state the equation for q \left(x\right), which is g \left( f \left(x^{2}\right)\right).

9

Consider the following functions:

\begin{aligned} f \left(x\right) &= x^{2} + 3 \\ g \left(x\right) &= 4 x - 9 \end{aligned}
a

State the equation for f \left( 2 x\right).

b

Show that f \left( 2 x\right) = g \left( f \left(x\right)\right).

10

The function f \left(x\right) is defined as f \left(x\right) = - 3 x + 4 g \left(x\right).

a

Given that f \left(x\right) is a quadratic function, what type of function is g \left(x\right)?

b

If f \left(x\right) = -3x + 20x^{2}, find the equation for g\left(x\right).

c

Find an algebraic expression for the function g \left( f \left(x\right)\right).

11

Consider the following functions:

\begin{aligned} p \left(x\right) &= x + 3 \\ q \left(x\right) &= x^{2} - 1 \\ r \left(x\right) &= \left(x + 4\right) \left(x + 2\right) \end{aligned}
a

Rewrite r \left(x\right) in expanded form.

b

Show that r \left(x\right) = q \left( p \left(x\right)\right).

c

Write an algebraic expression for p \left( q \left(x\right)\right).

d

Find the range of values for which r \left(x\right) < p \left( q \left(x\right)\right).

12

Consider the following functions:

\begin{aligned} f \left( x \right) &= \dfrac{4}{x} \\ g \left( x \right) &= x^{2} + 3 \end{aligned}

Find the following functions:

a

f \left( x \right) g \left( x \right)

b

f \left( g \left( x \right) \right)

c

\dfrac{f \left( x \right)}{g \left( x \right)}

d

\dfrac{g \left( x \right)}{f \left( x \right)}

13

Consider the following functions:

\begin{aligned} f \left( x \right) &= 9 x + 4 \\ g \left( x \right) &= x^{2} - 2 x - 3 \end{aligned}

Find the following functions:

a

f \left( x \right) g \left( x \right)

b

f \left( g \left( x \right) \right)

c

\dfrac{f \left( x \right)}{g \left( x \right)}

d

\dfrac{g \left( x \right)}{f \left( x \right)}

Domain and range
14

For each of the following functions f \left( x \right) and g \left( x \right):

i

Find the composite function f \left( g \left( x \right) \right).

ii

State the domain of f \left( g \left( x \right) \right).

iii

Find the composite function g \left( f \left( x \right) \right).

iv

State the domain of g \left( f \left( x \right) \right).

v

Find the composite function g^2 \left( x \right).

vi

State the domain of g^2 \left( x \right).

a

\begin{aligned} f \left( x \right) &= \dfrac{9}{1 - 2 x} \\ g \left( x \right) &= \dfrac{3}{x} \end{aligned}

b

\begin{aligned} f \left( x \right) &= \sqrt{x + 4} \\ g \left( x \right) &= x^{2} - 4 \end{aligned}

15

Consider the function h \left( x \right) = \sqrt{1 + \sqrt{1 + x}} and suppose that g \left( x \right) = \sqrt{1 + x}.

Find f \left( x \right) such that h(x) = f\left(g\left(x\right)\right).

16

Consider the function h \left( x \right) = \dfrac{1}{\left(x - 2\right)^{6}}.

a

If g \left( x \right) = x - 2, find f \left( x \right) such that h(x) = f\left(g\left(x\right)\right).

b

If f \left( x \right) = \dfrac{1}{x}, find g \left( x \right) such that h(x) = f\left(g\left(x\right)\right).

17

Consider the function f \left(x\right) = 5 x^{2} + 4. Define g \left(x\right) such that g \left( f \left(x\right)\right) = f \left(x\right) for all x.

18

For each of the following functions:

i

State the domain and range of f \left( x \right).

ii

State the domain and range of g \left( x \right).

iii

Hence, determine the domain and range of f \left( g \left( x \right) \right).

a

\begin{aligned}f \left( x \right) &= x^{2} \\ g \left( x \right) &= x - 7 \end{aligned}

b

\begin{aligned} f \left( x \right) &= \sqrt{x} \\ g \left( x \right) &= x - 3 \end{aligned}

c

\begin{aligned} f \left( x \right) &= \sqrt{x} \\ g \left( x \right) &= 4 x + 9 \end{aligned}

d

\begin{aligned} f \left( x \right) &= x^{3} \\ g \left( x \right) &= x + 8 \end{aligned}

19

Consider the following functions: \begin{aligned} f \left( x \right) &= \sqrt{x + 5} \\ g \left( x \right) &= 5 x - 6 \end{aligned}

a

Find an algebraic expression for the function f \left( g \left( x \right) \right).

b

Express the domain of f \left( g \left( x \right) \right), using inequalities.

c

Find an algebraic expression for the function g \left( f \left( x \right) \right).

d

Express the domain of g \left( f \left( x \right) \right), using inequalities.

20

Consider the following functions:

\begin{aligned} f \left( x \right) &= - 2 x \\ g \left( x \right) &= \dfrac{x}{x - 5} \end{aligned}
a

Find an algebraic expression for the function f \left( g \left( x \right) \right).

b

Determine the value(s) of x that are not in the domain of f \left( g \left( x \right) \right).

c

Find an algebraic expression for the function g \left( f \left( x \right) \right).

d

Determine the value(s) of x that are not in the domain of g \left( f \left( x \right) \right).

Applications
21

At sale time in a certain clothing store, all dresses are on sale for \$5 less than 75\% of the original price. a Write a function g\left(x\right) that finds 75\% of x. b Write a function f \left(x \right) that finds 5 less than x. c Construct the composite function f \left( g \left( x \right) \right). d Hence, calculate the sale price of a dress that has an original price of \$94.

22

A computer manufacturer sells hard drives to a retail outlet, each at a cost of \$11 more than the manufacturing cost. The retail store then sells each hard drive to the public, charging 60\% more than they paid to the manufacturer. a If m represents the manufacturing cost (in dollars), find a function A \left( m \right) which returns the price of buying a hard drive at the retail store. b Find the price of buying a hard drive at the store, if the manufacturing cost is \$22.50 .

### Outcomes

#### 0606C1.1

Understand the terms: function, domain, range (image set), one-one function, inverse function and composition of functions.

#### 0606C1.2A

Use function notation. e.g. f(x) = sin x, f: x ↦ sin x, f(x) = lg x or f: x ↦ lg x.

#### 0606C1.2C

Use the notation f^2(x) [= f(f(x))].

#### 0606C1.5B

Form composite functions.