2.01 Functions and relations

A relation is a set of ordered pairs which represent a relationship.

For example, we can think of the names of people in a math class and their ages as ordered pairs, like \left(\text{Bob}, 13 \right). These pairs of information represent a relation.

If we chose a specific age (like 13), we could list all the names of the people who are this age. It could be one person, Bob, or it could be multiple. If a teacher wanted to look for the person who was 13 years old, that description might fit four people which means there's not one clear answer.

We can express the same relation in several different ways: as a mapping, a set of ordered pairs, an input-output table, a graph in the coordinate plane, or as an equation in terms of x and/or y that describes a graph.

A mapping diagram shows how the input values are assigned one or more output values. Consider the mapping below:

\, \, \,

\, \, \,

A mapping of a relation

We can write an input-output table from the mapping, making sure that each pair is represented. Remember that the first value of a relation is an input value and the second value is the output value. The input is the value of x that is applied to the relation. The output is the y, or the answer that is received as a result of putting x into the relation. A table can be laid out horizontally (like the one shown below) or vertically.

This also corresponds to the set of ordered pairs \{(-1, 2), (0, 0), (1, 2), (2, 4)\}, which can be graphed in the coordinate plane, as shown below.

Examples

A function is a special type of relation where each input only has one output. Functions are a way of connecting input values to their corresponding output values.

For example, if we think about placing an order for boba teas, the number of boba teas we order (the input) affects the amount we have to pay (the output).

\, \\Let's say each boba tea costs \$3. If we bought one boba tea, it would cost \$3, if we bought two boba teas, it would cost \$6 and so on.

Notice how the value of the input (the number of boba teas) always produces exactly one output (cost)?

This is an example of a function.

\, \\However, if our friend also ordered 2 boba teas, but was charged \$10 while we paid \$6, we may be confused and assume that the cash register miscalculated the price.

This is not a function.

\, \\Another way to look at functions is to look at how inputs map to their output values. The function machine shown represents the relationship y=-2x.

Each input going into the machine is multiplied by -2 and exits the machine as the output, or y-value.

If we put four input values of -1,\,0,\,1,\, and 2, into the machine, it gives four output values of 2,\,0,\,-2,\, and -4.

Notice how each x-value is associated with only one y-value. This means y=-2x is a function.

If you can write a relationship between x and y then we can see that there is a relation. However, if this relationship only gives one value of y for each x-value (or one output for every input) then it is a function.

Sets of ordered pairs can also show whether or not a relation is a function. The following set of ordered pairs is a function since one input matches to one output:

\left\{\left(-4,-1\right),\left(-2,0\right),\left(0,1\right),\left(2,0\right),\left(3,-2\right),\left(4,3\right)\right\}

This set of ordered pairs is not a function since the inputs of -4 and 2 each repeat with different outputs.

\left\{\left(-4,-1\right),\left(-4,3\right),\left(0,1\right),\left(2,2\right),\left(2,-2\right),\left(4,0\right)\right\}

Graphs can also show one input (x-value) paired with one output (y-value ). By graphing the previous set of ordered pairs, we see the graph of a function and the graph not representing a function.

Examples

Example 4

This mapping shows the relation F.

a

Find the output when x=1.

Worked Solution

Create a strategy

We start from the input oval labeled x and follow the line(s) from 1 to the output value(s).

Apply the idea

When x=1, y=-1.

Watch question walkthrough

b

Determine if F is a function.

Worked Solution

Create a strategy

For a function, each x-value maps to a unique y-value.

Apply the idea

If we input x=2 into the mapping, we get both y=0 and y=1. This means that F is not a function.

Reflect and check

To be a function, each input value should only map to one output value. For example, G would be a function:

Watch question walkthrough

Example 5

The pairs of values in the table represent a relation between x and y.

x	-8	-7	-6	-3	2	7	9	9	10
y	8	13	-18	-16	-15	-2	-4	11	-9

Do they represent a function?

Worked Solution

Create a strategy

The relation is a function if for every x-value, there is exactly one y-value.

Apply the idea

The x-value of 9 yields the y-values of -4 and 11. So, the points do not represent a function.

Watch question walkthrough

Example 6

Oprah makes scarves to sell at the market. It costs her \$2 to produce each one, and she sells them for \$5.

a

Complete the graph of the points representing the relation between the number of scarves she manages to sell and her total profit for when 1, 2, 3, 4 and 5 scarves are sold. The first point has been plotted for you.

-8

-6

-4

-2

2

4

6

8

10

\text{Quantity}

2

4

6

8

10

12

14

16

18

\text{Profit}

Worked Solution

Create a strategy

The total profit can be found using the formula:

\text{Total profit}= \text{Total revenue}-\text{Total cost}

Apply the idea

\text{Total revenue}=\text{Number of scarves sold}\cdot \$5

\text{Total cost}=\text{Number of scarves sold}\cdot \$2

\displaystyle \text{Profit for 1 scarf}	\displaystyle =	\displaystyle 1\cdot 5 - 1\cdot 2	Substitute the number of scarves
	\displaystyle =	\displaystyle \$3	Evaluate

\displaystyle \text{Profit for 2 scarves}	\displaystyle =	\displaystyle 2\cdot 5 - 2\cdot 2	Substitute the number of scarves
	\displaystyle =	\displaystyle \$6	Evaluate

\displaystyle \text{Profit for 3 scarves}	\displaystyle =	\displaystyle 3\cdot 5 - 3\cdot 2	Substitute the number of scarves
	\displaystyle =	\displaystyle \$9	Evaluate

\displaystyle \text{Profit for 4 scarves}	\displaystyle =	\displaystyle 4\cdot 5 - 4\cdot 2	Substitute the number of scarves
	\displaystyle =	\displaystyle \$12	Evaluate

\displaystyle \text{Profit for 5 scarves}	\displaystyle =	\displaystyle 5\cdot 5 - 5\cdot 2	Substitute the number of scarves
	\displaystyle =	\displaystyle \$15	Evaluate

Plot the pairs of values found.

1

2

3

4

5

6

7

8

9

\text{Quantity}

2

4

6

8

10

12

14

16

18

\text{Profit}

Watch question walkthrough

b

Is this relation a function?

Worked Solution

Create a strategy

This relation is a function if for every quantity sold, there is exactly one total profit.

Apply the idea

Checking each pair of values in the graph, each quantity of scarves sold is associated with only one total profit, so this relation does represent a function.

Watch question walkthrough

Sometimes it is easier to investigate the graph of a relation to determine whether or not it is a function. When looking at a graph, if you can draw a vertical line anywhere so that it crosses the graph of the relation in more than one place, then it is not a function.

Here are two examples of relations being checked with the vertical line test. A function is said to "pass the vertical line test" while a relation that is not a function "fails the vertical line test."

One pair of points is enough to decide that a relation is not a function, but it is not enough to decide that a relation is a function. We must keep checking points on the graph until it either fails the test or we have checked for all x-values.

When classifying, remember that every function is a relation, but not every relation is a function.

Examples