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2.01 Functions and relations

Worksheet
Distinguishing between relations and functions
1

What is the name used to describe a graph where for some value of x, there exists two or more different values of y?

2

Determine whether the following statements are true or false:

a

When working with a function, substituting a certain value of x into the formula gives only 1 value of y for that value of x.

b

A horizontal line can intersect the graph of a function at more than one point.

c

A relation always passes the vertical line test.

d

All functions are relations.

3

State whether each of the following statements are correct:

a

Some relations are functions.

b

All functions are relations.

c

No functions are relations.

d

No relations are functions,

4

State whether each of the following statements are correct:

a

The graph of every non-vertical straight line is a function.

b

There is no straight line graph that is a function.

c

The graph of every straight line is a relation.

d

The graph of every non-horizontal straight line is a function.

5

Determine whether each of the following are:

  • a function,

  • a relation,

  • a relation and a function, or

  • neither

a
-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
b
-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
c
-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
d
-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
e
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
f
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
g
-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
h
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
i
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
j
-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
k

\left\{\left(2, 5\right), \left(7, - 3 \right), \left(5, 2\right), \left( - 4 , - 9 \right)\right\}

l

\left\{\left(2, 5\right), \left(2, 7\right), \left( - 3 , - 4 \right), \left( - 9 , 13\right)\right\}

m

(1,5), (1,1), (7,-2), (-5,-10)

n

(1,5), (7,-2), (-5,-10), (13,-13)

o

(1,5), (1,7), (-2,-5), (-5,-10)

p

y = 9 x

q

y = x^{2} + 2

r

y = \dfrac{3}{x}

s

y^{2} + x^{2} = 25

t

y^{2} = x

u

y = - x^{3}

6

State whether the values in the tables below represent a function:

a
x-4-3-2-101234
y-4-3-2-10-1-2-3-4
b
x0148912161820
y012\dfrac{2}{\sqrt{2}}3\dfrac{2}{\sqrt{3}}4\dfrac{3}{\sqrt{2}}\dfrac{2}{\sqrt{5}}
c
x-9-7-6-5-3-23510
y101010101010101010
d
x-3-2-20112
y8437210
7

Consider the points in the table.

x-4-3-2-101234
y432101234
a

Plot the points on a number plane.

b

Do they represent a function?

c

Write down the coordinates of all the points that would need to be removed so that the remaining points form an increasing function.

8

Consider the points in the table:

x-4-30-101234
y432101234
a

Plot the points on a number plane.

b

Do they represent a function?

9

A relation is defined as follows:

y = 1 if x is positive and y = -1 if x is 0 or negative.

a

Complete the table for this relation.

x-4-3-2-101234
y
b

Plot the points on a number plane.

c

Do these values represent a function?

10

Consider these ordered pairs:

\left\{\left( - 9 , - 5 \right), \left( - 5 , - 10 \right), \left( - 5 , - 4 \right), \left( - 3 , 7\right), \left( - 2 , - 4 \right), \left( - 1 , 1\right)\right\}

a

Plot the ordered pairs on a number plane.

b

Which ordered pair would need to be removed from the set so that the remaining ordered pairs represent a function?

11

Write down a value of k such that the relation \left\{\left(6, 2\right), \left(8, 5\right), \left(1, 7\right), \left(k, 4\right)\right\} does not represent a function.

12

Does there exist any values of k such that the relation \left\{\left(2, 1\right), \left(7, k\right), \left(5, 8\right), \left(3, 9\right)\right\} does not represent a function?

Applications
13

CheapCalls Mobile charges \$1.10 a minute plus a connection fee of 70c for an international call.

a

Complete the given table:

b

Is this relation a function?

Call length (minutes)International Call cost (dollars)
1
2
3
4
5
14

A particular store is offering one free T-shirt for every two T-shirts purchased, where each T-shirt costs \$19.

a

Complete the given table:

b

Is this relation a function?

Number of T-shirtsTotal Cost (dollars)
1
2
3
4
5
15

A particular internet service provider charges \$40 a month on their 20 GB plan and charges \$10 for each additional 10 GB used.

a

Complete the given table:

b

Is this relation a function?

Total GB UsedTotal Charge (dollars)
10
20
30
40
50
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Outcomes

1.2.1.1

understand the concept of a relation as a mapping between sets, a graph and as a rule or a formula that defines one variable quantity in terms of another

1.2.1.2

recognise the distinction between functions and relations and use the vertical line test to determine whether a relation is a function

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