Video Transcript
Which of the following relations
represents a function?
In this question, we are given two
relations 𝐴 and 𝐵, both of which consist of five ordered pairs. We begin by recalling that a
function is a rule that takes every member of a set and maps it onto exactly one
member in a second set. For a group of ordered pairs 𝑥,
𝑦, the input is 𝑥 and the output is 𝑦. This means that for a group of
ordered pairs to represent a function, no two ordered pairs can have the same input
with a different output. In other words, if an ordered pair
shares the same 𝑥-value, their 𝑦-value must also be the same.
We observe that relation 𝐴 has two
ordered pairs with an 𝑥-value of four. For relation 𝐴 to be a function,
the corresponding 𝑦-values in these ordered pairs must also be the same. However, the ordered pairs are
four, 12 and four, 15. Since the 𝑦-values are not the
same, relation 𝐴 cannot represent a function. Relation 𝐴 also has two ordered
pairs with an 𝑥-value of five. They too have different 𝑦-values:
the ordered pairs five, 18 and five, 21. We can therefore conclude that
relation 𝐴 does not represent a function.
The ordered pairs of relation 𝐵
have unique 𝑥-values: the integers four, five, six, seven, and eight. This satisfies the condition that a
function takes every member of a set and maps it onto exactly one member in the
second set. We can therefore conclude that the
relation that represents a function is relation 𝐵.
We could also represent these
ordered pairs on a mapping diagram. If we begin by considering relation
𝐴, we observe that elements four and five from the input or 𝑥-column map to more
than one element in the 𝑦- or output column. This mapping diagram therefore
cannot represent a function. For a mapping diagram to represent
a function, every input value must have a single output. When we consider relation 𝐵, we
see that every input, the integers four, five, six, seven, and eight, have a single
output, the integers 12, 15, 18, 21, and 24. Whilst it is not required for this
question, we notice that the 𝑦-values are three times the 𝑥-values. The function can therefore be
written as the equation 𝑦 is equal to three 𝑥.
