### Video Transcript

Which of the following correctly
express the relation 𝑅 illustrated in the figure below?

Any relation is a set of ordered
pairs of the form 𝑥, 𝑦. The set of 𝑥-values is known as
the domain or input, and the 𝑦-values are the range or output. In terms of a mapping diagram like
this, the 𝑥-values will be where the arrows start and the 𝑦-values will be where
the arrows finish. We could go straight to our diagram
and list the correct ordered pairs. Alternatively, we could look at the
five options and eliminate some immediately.

Option (A) 𝑅 is equal to the set
of negative 18, negative nine, zero, nine, and 18. This is just a set of values and
not a set of ordered pairs. Therefore, option (A) cannot be
correct. Option (B) 𝑅 is equal to negative
18, 18 and negative nine, nine. As there are five arrows on the
figure, we know there must be five ordered pairs. This means that option (B) cannot
be correct. Option (C) has four ordered pairs:
negative 18, 18; negative nine, nine; nine, negative nine; and 18, negative 18. This means that this option is also
incorrect.

Both options (D) and (E) contain
five ordered pairs. However, option (E) contains
fractional values: negative one eighteenth, negative one-ninth, one-ninth, and one
eighteenth. As these values do not appear on
the figure, this answer is also incorrect. By elimination, we have found that
option (D) must be the correct answer — negative 18, 18; negative nine, nine; zero,
zero; nine, negative nine; and 18, negative 18. We’ll now check that these are
indeed the five ordered pairs on the figure. The arrow that starts furthest to
the left starts at negative 18 and finishes at 18. This means it has an input of
negative 18 and an output of 18 and is therefore the ordered pair negative 18,
18.

The second arrow along goes from
negative nine to nine. This is therefore another ordered
pair. The third arrow starts at zero but
also does a loop and goes back to zero. This corresponds to the ordered
pair zero, zero. Our next arrow has an input of nine
and an output of negative nine. This is the ordered pair nine,
negative nine. Finally, we have an arrow that
starts at 18 and finishes at negative 18. The five ordered pairs are negative
18, 18; negative nine, nine; zero, zero; nine, negative nine; and 18, negative
18. This confirms that option (D) is
correct.

We might notice a pattern here
between our input and output values. If we multiply or divide each of
the input values by negative one, we get the output values. This means that this mapping
corresponds to the function 𝑦 is equal to negative 𝑥.