# 3.03 Identifying functions

Lesson

We just learned that a relation is a relationship between sets of information. Recall the example of the people in a math class and their heights, where someone was looking for the person who was $162$162 cm tall. We realized that description might fit multiple people! There's not one clear answer. This data could be expressed as a relation.

### What is a function?

A function is a special type of relation, where each input only has one output. We are already familiar with inputs and outputs. Functions are a way or connecting input values to their corresponding output values. For example, if we think about placing an order for smoothies, the number of smoothies we order (the input) affects the amount we have to pay (the output). Let's say each smoothie costs $\$3$$3. If we bought one smoothie, it would cost \3$$3, if we bought two smoothies, it would cost $\$6$$6 and so on. Do you notice how the value of our input (the number of smoothies) always produces exactly one output (cost)? This is an example of a function. Let's look at another example. Say we have the expression y=2xy=2x. Let's construct a table of values to record the results. Based on the equation we can find the yy-value by multiplying the xx-value by 22. xx -11 00 11 22 yy -22 00 22 44 See how each xx value is associated with only one yy value? This means this data displays a function. ### Identifying functions If you can write a relationship between xx and yy then we can see that there is a relation. However, if this relationship only yields one value of yy for each xx value (or one output for every input) then it is a function. #### Worked Examples ##### Question 1 Consider the relation shown in the table below. Does it represent a function? xx yy 22 44 66 99 33 1212 22 77 Think: Remember, for a relation to be considered a function, every xx-value should yield only one yy-value. Do: Look for any xx-values that yield more than one yy-value. Notice that the xx-value of 22 yields the yy-values of 44 and 77, therefore this table does not represent a function. ##### Question 2 Determine if the following relation is a function: \left\{\left(-2,3\right),\left(-1,5\right),\left(0,7\right),\left(1,9\right),\left(2,11\right)\right\}{(2,3),(1,5),(0,7),(1,9),(2,11)} Think: Recall that ordered pairs take the form x,yx,y so check to make sure that no xx-coordinates share a common yy-coordinate. Do: In this case, each xx-coordinate is associated with only one yy-coordinate so this relation does represent a function. ### The vertical line test 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 is an example of a relation that is not a function. See how when the blue vertical line is drawn in, it crossed the graph in two places? Careful! Make sure to check the entire graph. In other words, functions have to pass the vertical line test at every point. If it fails in even one spot then it is not a function. ##### Question 3 Determine whether the following graph describes a function or a relation. Think: Does this graph pass the vertical line test at every point? Do: If a vertical line is drawn at any point on the graph, it only crosses through one point at a time. Reflect: This graph does represent a function. Remember! While all functions are relations, not all relations are functions. #### Practice questions ##### Question 4 The pairs of values in the table represent a relation between xx and yy. Do they represent a function?  xx yy -8−8 -7−7 -6−6 -3−3 22 77 99 99 1010 88 1313 -18−18 -16−16 -15−15 -2−2 -4−4 1111 -9−9 1. Yes A No B Yes A No B ##### Question 5 Tracy makes scarfs to sell at the market. It costs her \3$$3 to produce each one, and she sells them for $\$66.

1. Consider when $1$1, $2$2, $3$3, $4$4 and $5$5 scarfs are sold. Graph the points representing the relation between the number of scarfs she manages to sell and her total profit.

2. Is this relation a function?

No

A

Yes

B

No

A

Yes

B

### Outcomes

#### 8.15a

Determine whether a given relation is a function