iGCSE (2021 Edition)

# 4.03 Inverse functions

Worksheet
Inverse functions
1

Explain how a graphing utility such as a graphing calculator be used to visually determine if two functions are inverses of each other.

2

In each graph below two functions and the line y = x are drawn. For each graph, state whether the functions are inverse functions of each other.

a
b
c
d
e
f
g
3

For each of the following functions, find:

i
f^{-1}(x)
ii

The domain of f^{-1}(x)

iii

The range of f^{-1}(x)

a

f \left( x \right) = 8 x - 9 defined over \left[ - 4 , 2\right]

b

f \left( x \right) = x^{2} defined over \left[0, \infty\right)

c

f \left( x \right) = \sqrt{16 - x^{2}} defined over \left[0, 4\right]

4

State whether the following functions have an inverse function:

a

f(x) = \dfrac{7}{x}

b

f(x) = \sqrt{7 - x^{2}}

c

f(x) = \sqrt{x} + 7

d

f(x) = 1 - 7 x

e

f(x) = 7^{x}

f

f(x) = 4 x^{3} + 8

g

f(x) = \sqrt{4 + x}

h

f(x) = - \dfrac{1}{7} x^{2}

5

State whether the following functions have an inverse function:

a
b
c
d
e
Graphs of inverse functions
6

Consider the functions f \left( x \right) = \dfrac{1}{x} + 3 and g \left( x \right) = \dfrac{1}{x - 3}.

a

Sketch the graph of f \left( x \right).

b

Sketch the graph of g \left( x \right) on the same set of axes.

c

Are f \left( x \right) and g \left( x \right) inverses?

7

Consider the graph of the function f \left( x \right) over the line y = x:

Sketch the graph of f^{ - 1 } \left(x\right).

8

Consider the graphs of f \left( x \right) and g \left( x \right):

a

State the equation of f \left( x \right).

b

State the equation of g \left( x \right).

c

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

d

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

e

State whether the following statements are correct:

i

g \left( x \right) is an inverse of f \left( x \right).

ii

f \left( g \left( x \right) \right) has gradient - 2.

iii

f \left( x \right) is an inverse of g \left( x \right).

iv

g \left( f \left( x \right) \right) has gradient 1.

9

Consider the graph of y = \dfrac{2}{x} over the line y = x:

a

Sketch the graph of the inverse of \\y = \dfrac{2}{x}.

b

Compare the inverse graph to the original graph.

10

Sketch the graph of the inverse of the following functions:

a
b
c
d
e
f
g
h
i
j
11

For each of the following functions:

i

Sketch the function f \left( x \right) over its domain.

ii

Find the inverse, f ^{-1}.

iii

State the domain of f ^{-1}.

iii

State the range of f ^{-1}.

iv

Sketch the function f ^{-1} over its domain.

a

f \left( x \right) = x + 6 defined over the interval \left[0, \infty\right).

b

f \left( x \right) = 7 - x defined over the interval \left[2, 9\right].

c

f \left( x \right) = \left(x - 6\right)^{2} - 2 defined over the interval \left[6, \infty\right).

d

f \left( x \right) = \sqrt{4 - x} defined over the interval \left[0, 4\right).

e

f \left( x \right) = \left(x + 2\right)^{2} + 3 defined over the interval \left[0, \infty\right).

12

Consider the functions f \left( x \right) = x^{2} - 5 and g \left( x \right) = \sqrt{x + 5}, for x \geq 0. The function y is defined as y = g \left( f \left( x \right) \right), for x \geq 0.

a

State the equation for y.

b

Graph the functions f \left( x \right), g \left( x \right) and y on the same set of axes.

c

What do you notice about the graph of y in relation to the graphs of f \left( x \right) and g \left( x \right)?

Applications
13

The function t = \sqrt{\dfrac{d}{4.9}} can be used to find the number of seconds it takes for an object in Earth's atmosphere to fall d metres.

a

State the function for d in terms of t.

b

Find the distance a skydiver has fallen 5 seconds after jumping out of a plane.

14

The following formula can be used to convert Fahrenheit temperatures x to Celsius temperatures T \left( x \right):

T \left( x \right) = \dfrac{5}{9} \left(x - 32\right)

a

Find T \left( - 13 \right).

b

Find T \left( 86 \right).

c

Find T^{ - 1 } \left(x\right).

d

What can the formula T^{ - 1 } be used for?

15

The function d\left(t\right) = 120 - 4.9 t^{2} can be used to find the distance, d, that an object dropped from a height of 120 metres has fallen after t seconds.

a

Is the function d \left(t\right) one-to-one?

b

Find the inverse function, t \left(d\right).

c

Prove that d \left( t \left(d\right)\right) = d.

d

Prove that t \left( d \left(t\right)\right) = t.

e

Are d and t inverse functions?

f

How long it will take an object to fall 41.6 metres when dropped from a height of 120 metres?

16
a

Consider the function f \left( x \right) = \left(x - 2\right)^{2} - 6. Write the function as two one-to-one functions that have the same rule as f \left( x \right), but specific domains.

b

Consider the graph of y = g \left( x \right) shown:

Complete the piecewise function h(x) below to split the graph of y = g \left( x \right) into three one-to-one functions that have the same rule as g \left( x \right):

h(x) = \begin{cases} g(x), & x\leq ⬚ \\ g(x), & ⬚ \lt x \lt ⬚ \\ g(x), & x \geq ⬚ \end{cases}

### Outcomes

#### 0606C1.1

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

#### 0606C1.2B

Use the notation f ^(–1)(x).

#### 0606C1.2C

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

#### 0606C1.4

Explain in words why a given function is a function or why it does not have an inverse.

#### 0606C1.5A

Find the inverse of a one-one function.

#### 0606C1.5B

Form composite functions.

#### 0606C1.6

Sketch graphs to show the relationship between a function and its inverse.