5.04 Domain and range
Functions and relations have been described as ways of linking two sets of data, usually written using $x$x's and $y$y's as coordinates. We also learnt that relations can only be functions if there's only one $y$y value associated with each $x$x value.
For example, $\left\{\left(1,2\right),\left(5,3\right),\left(2,-7\right),\left(5,-1\right)\right\}${(1,2),(5,3),(2,−7),(5,−1)} is not a function as the same $x$x value of $5$5 can be related to the $y$y values of both $3$3 and $-1$−1. We also know that graphically, we can use the vertical line test to see if a relation is a function.
A relation where there is more than one $x$x value with the same $y$y value can still be a function! Do not confuse this with the criteria for a function: only one $y$y value for each $x$x value.
Domain and range
These two words sound similar, don't they? They both seem to be talking about areas and spans, and in maths they have similar definitions:
Domain - the possible $x$x values of a relation
Range - the possible $y$y values of a relation
There are a number of ways to find the domain and range of a relation.
Method 1: Look at the coordinates given and simply list the possible values. In the previous example, the domain is $\left\{1,5,2\right\}${1,5,2} and the range is $\left\{2,3,-7,-1\right\}${2,3,−7,−1}. Notice how repeated values are not included and order is not important, as we only care about the POSSIBLE values of $x$x and $y$y.
Method 2: Look at a relation graphically, and see how 'wide' or 'long' the graph is:
Horizontally this graph spans from $-1$−1 to $1$1, so we can write the domain as $-1\le x\le1$−1≤x≤1. Similarly, the graph goes vertically from $-2$−2 to $2$2 so the range can be written as $-2\le y\le2$−2≤y≤2.
Practice questions
Question 1
Consider the relation in the table.
| $x$x | $y$y |
|---|---|
| $1$1 | $3$3 |
| $6$6 | $2$2 |
| $3$3 | $7$7 |
| $8$8 | $1$1 |
| $2$2 | $2$2 |
What is the domain of the relation? Enter the values, separated by commas.
What is the range of the relation? Enter the values separated by commas.
Is this relation a function?
Yes
ANo
B
question 2
Consider the graph of the relation on the $xy$xy-plane below.
What is the domain of the relation?
Express your answer using inequalities.
What is the range of the relation?
Express your answer using inequalities.
Is this relation a function?
Yes
ANo
B
Worked example
A cafe is running a promotion where every $3$3 cups of coffee earns you a free cup, and each cup costs $\$3.50$$3.50.
(a) Make a table of prices for $1$1 to $8$8 cups of coffee
Think: Which pairs of numbers of cups would be equal to the same price?
Do:
$3$3 cups and $4$4 cups would cost the same as the $4$4th cup would be free, same goes for $7$7 and $8$8 cups. Therefore:
| Number of cups | Price ($) |
|---|---|
| $1$1 | $3.50$3.50 |
| $2$2 | $7$7 |
| $3$3 | $10.50$10.50 |
| $4$4 | $10.50$10.50 |
| $5$5 | $14$14 |
| $6$6 | $17.50$17.50 |
| $7$7 | $21$21 |
| $8$8 | $21$21 |
(b) Determine: Is this relation a function?
Think: In word problems, $x$x usually represents the independent variable and $y$y the dependent variable. Are there any $x$x-values here with more than one $y$y-value?
Do:
The left column can be represented by $x$x and the right by $y$y. We can see that we can some doubling up of $y$y-values for different $x$x-values but for each $x$x-value there is only one $y$y-value. Therefore this is a function.