New Zealand
Level 7 - NCEA Level 2

# Domain and Range

Lesson

Functions and relations have been explored by us in the past to describe 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.

Careful!

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:

Remember!

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. One is to look at the coordinates given and simply list the possible values. For example 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

The other method is to 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$1x1. Similarly, the graph goes vertically from $-2$2 to $2$2 so the range can be written as $-2\le y\le2$2y2.

#### Examples

##### 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
1. What is the domain of the relation? Enter the values, separated by commas.

2. What is the range of the relation? Enter the values separated by commas.

3. Is this relation a function?

Yes

A

No

B

Yes

A

No

B

##### question 2

Consider the graph of the relation on the $xy$xy-plane below.

1. What is the domain of the relation?

2. What is the range of the relation?

3. Is this relation a function?

Yes

A

No

B

Yes

A

No

B

##### question 3

A cafe is running a promotion where every $3$3 cups of coffee earns you a free cup, and each cup costs $\$3.503.50

a) Draw: 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.

### Outcomes

#### M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

#### 91257

Apply graphical methods in solving problems