Suppose each student in your class were represented by a point, with the \begin{align*}x\end{align*}-coordinate being the student's year in school and the \begin{align*}y\end{align*}-coordinate being the student's age. Would the set of points represent a function? How would you know? What if you graphed the set of points on a Cartesian plane? Is there any sort of test you could use to find out if you had a function?

### The Vertical Line Test

You saw that a function is a **relation** between the independent and the dependent variables. It is a rule that uses the values of the independent variable to give the values of the dependent variable. A function rule can be expressed in words, as an equation, as a table of values, and as a graph. All representations are useful and necessary in understanding the relation between the variables.

One way to determine whether a relation is a function is to construct a **flow chart** linking each dependent value to its matching independent value. Consider the relation that shows the heights of all students in a class. The domain is the set of people in the class and the range is the set of heights. Each person in the class cannot be more than one height at the same time. This relation is a function because for each person there is exactly one height that belongs to him or her.

Notice that in a function, a value in the range can belong to more than one element in the domain, so more than one person in the class can have the same height. The opposite is not possible; that is, one person cannot have multiple heights.

#### Let's determine if the following relations are functions:

- (1, 3), (–1, –2), (3, 5), (2, 5), (3, 4)

To determine whether this relation is a function, we must use the definition of a function. Each \begin{align*}x-\end{align*}coordinate can have ONLY one \begin{align*}y-\end{align*}coordinate. However, since the \begin{align*}x-\end{align*}coordinate of 3 has two \begin{align*}y-\end{align*}coordinates, 4 and 5, this relation is NOT a function.

- (–3, 20), (–5, 25), (–1, 5), (7, 12), (9, 2)

Applying the definition of a function, each \begin{align*}x-\end{align*}coordinate has only one \begin{align*}y-\end{align*}coordinate. Therefore, this relation is a function.

**Determining Whether a Graph Is a Function**

Suppose all you are given is the graph of the relation. How can you determine whether it is a function?

You could organize the ordered pairs into a table or a flow chart, similar to the student and height situation. This could be a lengthy process, but it is one possible way. A second way is to use the **vertical line test.** Applying this test gives a quick and effective visual to decide if the graph is a function.

A relation is a function if there are no vertical lines that intersect the graphed relation in more than one point. If a graphed relation does not intersect a vertical line in more than one point, then that relation is a function.

#### Now, let's determine if the following graphs are functions:

By drawing a vertical line (the red line) through the graph, we can see that the vertical line intersects the circle more than once. Therefore, this graph is NOT a function.

No matter where a vertical line is drawn through the graph, there will be only one intersection. Therefore, this graph is a function.

### Examples

#### Example 1

Earlier, you were asked if the set of points with the \begin{align*}x\end{align*}-coordinate being the student's year in school and the \begin{align*}y\end{align*}-coordinate being the student's age would represent a function. How would you know?

This would not be a function since there may be students in the same grade who are different ages. For example, a 8th grade student could be 13 or 14 depending on when their birthday is. If you were to graph this, there would be the two points (8, 13) and (8, 14). These two points would not pass the vertical line test because the same \begin{align*}x\end{align*} has two different \begin{align*}y\end{align*} values. If you have a graph, always remember that the vertical line test can be used to determine if a relation is a function.

#### Example 2

Determine if the graphed relation is a function:

Imagine moving a vertical line across the plane. Do you see anywhere that this vertical line would intersect the graph at more than one place?

There is no place on this graph where a vertical line would intersect the graph at more than one place. Using the vertical line test, we can conclude the relation is a function.

### Review

In 1-4, determine if the relation is a function.

- (1, 7), (2, 7), (3, 8), (4, 8), (5, 9)
- (1, 1), (1, –1), (4, 2), (4, –2), (9, 3), (9, –3)

- \begin{align*}&\text{Age} && 20 && 25 && 25 && 30 && 35\\ &\text{Number of jobs by that age} && 3 && 4 && 7 && 4 && 2\end{align*}
- \begin{align*}&& x && -4 && -3 && -2 && -1 && 0\\ && y && 16 && 9 && 4 && 1 && 0\end{align*}

In 5-6, write a function rule for the graphed relation.

In 7-8, determine whether the graphed relation is a function.

**Mixed Review**

- A theme park charges $12 entry to visitors. Find the money taken if 1296 people visit the park.
- A group of students are in a room. After 25 students leave, it is found that \begin{align*}\frac{2}{3}\end{align*} of the original group are left in the room. How many students were in the room at the start?
- Evaluate the expression \begin{align*}\frac{x^2+9}{y+2}\end{align*} when \begin{align*}y = 3\end{align*}and\begin{align*}x=4\end{align*}.
- The amount of rubber needed to make a playground ball is found by the formula \begin{align*}A = 4 \pi r^2\end{align*}, where \begin{align*}r=radius\end{align*}. Determine the amount of material needed to make a ball with a 7-inch radius.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.14.