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5.03 Identifying functions


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 $y=2x$y=2x. Let's construct a table of values to record the results. Based on the equation we can find the $y$y-value by multiplying the $x$x-value by $2$2.

$x$x $-1$1 $0$0 $1$1 $2$2
$y$y $-2$2 $0$0 $2$2 $4$4

See how each $x$x value is associated with only one $y$y value? This means this data displays a function.


Identifying functions

If you can write a relationship between $x$x and $y$y then we can see that there is a relation. However, if this relationship only yields one value of $y$y for each $x$x 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?

$x$x $y$y
$2$2 $4$4
$6$6 $9$9
$3$3 $12$12
$2$2 $7$7

Think: Remember, for a relation to be considered a function, every $x$x-value should yield only one $y$y-value.

Do: Look for any $x$x-values that yield more than one $y$y-value. Notice that the $x$x-value of $2$2 yields the $y$y-values of $4$4 and $7$7, 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 $x,y$x,y so check to make sure that no $x$x-coordinates share a common $y$y-coordinate.

Do: In this case, each $x$x-coordinate is associated with only one $y$y-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?



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.


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 $x$x and $y$y. Do they represent a function?

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








Question 5

Tracy makes scarfs to sell at the market. It costs her $\$3$$3 to produce each one, and she sells them for $\$6$$6.

  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.

    Loading Graph...

  2. Is this relation a function?











Understand that a function assigns to each x-value (independent variable) exactly one y-value (dependent variable), and that the graph of a function is the set of ordered pairs (x,y).

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