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Standard Level

5.04 Domain and range

Worksheet
Domain and range
1

For each of the following relations:

i

Find the domain.

ii

Find the range.

iii

Determine whether the relation is a function or not.

a

\left\{\left(8, 5\right), \left(1, 3\right), \left(3, 4\right), \left(9, 1\right), \left(2, 9\right)\right\}

b

\left\{\left(2, 3\right), \left(3, 8\right), \left(6, 1\right), \left(8, 7\right), \left(3, 2\right)\right\}

c

\left\{\left(4, 4\right), \left(8, 9\right), \left(2, 8\right), \left(7, 9\right), \left(3, 1\right)\right\}

d
x16382
y32712
e
x77853
y19326
f
2
4
6
8
10
12
x
2
4
6
8
10
12
y
g
2
4
6
8
x
2
4
6
8
10
12
y
h
2
4
6
8
10
x
2
4
6
y
i
2
4
6
8
10
x
2
4
6
8
y
j
2
4
6
8
10
12
x
2
4
6
8
10
12
y
k
-10
-5
5
10
x
-10
-5
5
10
y
l
-10
-5
5
10
x
-10
-5
5
10
y
m
-10
-5
5
10
x
5
10
y
n
-5
5
x
-5
5
y
2

State the domain and range of the following functions:

a
-7
-6
-5
-4
-3
-2
-1
1
2
x
-7
-6
-5
-4
-3
-2
-1
1
2
y
b
-7
-6
-5
-4
-3
-2
-1
1
2
x
-2
-1
1
2
3
4
5
6
7
y
c
-2
-1
1
2
3
4
5
6
7
x
-2
-1
1
2
3
4
5
6
7
y
3

Consider the function f graphed. Use the graph to determine whether each of the given statements are true or false:

a

The domain is \left[ - 5 , 7\right]

b

The range is \left[ - 3 , 16\right]

c

f \left( 1 \right) - f \left( 7 \right) = 16

d

f \left( 0 \right) = 16

-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
x
-4
-2
2
4
6
8
10
12
14
16
y
4

What is the domain of the following functions?

a

f \left( x \right) = - 3 x - 1

b

f \left( x \right) = 8 x + 8

c
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y
5

The graph of y = 2^{x} is shown below:

a

What is the y-intercept of this graph?

b

Does the graph have an x-intercept?

c

What is the graph's domain?

d

What is the graph's range?

e

Find the value of y when x = 7.

f

Find the value of x when y = 256.

-5
-4
-3
-2
-1
1
2
3
4
5
x
1
2
3
4
5
6
7
y
6

Consider the graph of the function shown:

a

What is the maximum value of the graph?

b

Hence, determine the range of the function.

c

What is the domain of this function?

-9
-8
-7
-6
-5
-4
-3
-2
-1
1
x
-4
-2
2
4
6
8
10
12
14
16
y
7

Consider the graph of the function shown:

a

What is the minimum value of the graph?

b

Hence, determine the range of the function.

c

Over what interval of the domain is the function increasing?

2
4
6
8
10
12
14
x
10
20
30
40
50
60
70
80
90
y
8

For the following quadratic functions:

i
Determine whether the function has a maximum or a minimum point.
ii
Find the minimum or maximum value of the function.
iii
Find the range of the function.
a
xy
117
27
31
4-1
51
67
717
b
xy
-5.5-12
-5-7
-4.5-4
-4-3
-3.5-4
-3-7
-2.5-12
9

Consider the graph of the function f \left(x\right) = \sqrt{x + 1} below:

a

State the domain of the function.

b

Is there a value in the domain that can produce a function value of -2?

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
10

Consider the parabola defined by the equation y = x^{2} + 5.

a

Is the parabola concave up or concave down?

b

What is the minimum y value of the parabola?

c

Hence, determine the range of the parabola.

Graphing functions
11

For the following functions:

i
Sketch the graph of the function.
ii
Find the domain.
iii
Find the range.
a

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

b

f \left( x \right) = 3 x - 1

c

f \left( x \right) = - \dfrac{3}{2} x - 2

d

f \left( x \right) = - 4

e

x = - 1

12

Consider the function y = x^{3}-1.

a

Complete the table of values:

x-2-1 012
y
b

Sketch the graph of the function.

c

Is there a value in the domain that can produce a function value of 3 \dfrac{2}{3} ?

13

Consider f \left(x\right) = x + 3 for the domain \{- 5, -4, 0, 1\}.

a

Complete the table of values:

x-5-401
f(x)
b

Plot the points on a number plane.

14

Consider f \left(x\right) = - 2 x for the domain \{- 1, 0, 1, 2\}.

a

Complete the table of values:

x-1012
f(x)
b

Plot the points on a number plane.

15

Consider the function f \left(x\right) = \dfrac{x + 3}{2}, for all real x.

a

Complete the table of values:

x34567
y
b

Sketch the graph of the function.

16

Consider - 2 x + y = 1 for the domain \{- 4, -3, 0, 2\}.

a

Make y the subject of the equation.

b

Complete the table of values:

x-4-302
y
c

Plot the points on a number plane.

17

Consider f \left(x\right) = x^{2} + 4 for the domain \{-3, -1, 0, 1, 3\}.

a

Complete the table of values:

x-3-1013
f(x)
b

Plot the points on a number plane.

18

Consider f \left(x\right) = \left(x + 2\right)^{2} for the domain \{- 2, -1, 0, 1, 2\}.

a

Complete the table of values:

x-2-1012
f(x)
b

Plot the points on a number plane.

19

The function f is used to determine the area of a square given its side length.

a

State whether the following values are part of the domain of the function:

i

- 8

ii

6.5

iii

9\dfrac{1}{3}

iv

\sqrt{78}

b

For n \geq 0, state the area function for a side length of n.

c

Sketch the graph of the function f.

Rational functions
20

What is the domain of the function defined by f \left( x \right) = \dfrac{1}{x + 5} ?

21

For each of the following graphs of rational functions:

i

State the domain using interval notation.

ii

State the range using interval notation.

a
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
b
-12
-10
-8
-6
-4
-2
2
4
6
8
10
12
x
-8
-6
-4
-2
2
4
6
8
y
Applications
22

An industrial process involves heating a material from 25°C to 80°C in order to remove impurities.

Using the standard approach, the temperature of the material over time is given by the function: T = -t \left(t - 16\right) + 25

A graph of the temperature is given, where temperature T is given in degrees Celsius(°C) and time t is in minutes.

a

What is the domain of this function?

b

What is the range of this function?

2
4
6
8
10
12
14
16
18
t
10
20
30
40
50
60
70
80
90
T
c

A new heating process is being developed that uses more energy but takes less time. This regime is based on a different quadratic function, and the table below shows the theoretical value of the temperature at different points in time using the new function.

t0134303132333435
T2545809696806345254

For what values of t does this new function have the same range as the standard process?

d

Hence, what values of t should be used for the new heating process?

23

The number of bees in a colony is initially measured and left to form a hive. The number of bees in the hive is measured each day over the course of one week. The function

H \left(x\right) = 30 \left(3\right)^{x} is found to model the number of bees, H, after x days.

a

What is the domain of the function?

b

What is the initial number of bees in the colony?

c

Can the function be used to model the number of bees over an entire season?

24

At an indoor ski facility, the temperature is set to - 8°C at the opening time of 9 am. At 10 am, the temperature is immediately brought down to - 15°C and left for 3 hours before immediately taking it down again to - 23 °C where it stays until the close time of 7 pm.

a

Write the piecewise function that models the indoor temperature f\left(x\right) in terms of the number of hours after the facility has opened, x.

b

The graph of the function is given. State the domain of the function.

c

Kate entered the ski facility at 2:30 pm. What was the temperature inside the facility?

d

Vincent wants to wait until the indoor temperature is - 10\degree C or lower. When is the earliest he can enter the facility?

1
2
3
4
5
6
7
8
9
10
x
-22
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
y
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