On a road trip, you stop at a vending machine during a lunch break. Each item in the vending machine has a unique code that consists of a letter followed by a number. No two items in the machine are the same. Is the vending machine an example of a function?

### Relations and Functions

Functions are a very important part of Algebra II. From this point on, we are going to study several different types of functions: linear, quadratic, cubic, polynomial, rational and trigonometric.

First, every set of points is called a **relation.** A relation is a grouped set of points that relate, or have something in common with each other. Here are a few examples of relations.

Whenever we talk about a set of points, the { }, or brackets are used. In the examples above, there are two sets of points. All 6 of these sets or equations are examples of relations. Relations also have an **input** and an **output.** Typically, all inputs are the **domain**), and all outputs are the **range**). The input could also be considered the **independent variable** and the output would be the **dependent variable.** Again, the

A more specific type of relation is a function. A **function** is a relation where there is exactly one output for every input. It cannot be a function if at least one input has more than one output. Simply stated, the **function notation.**

Let's compare {(3, -2), (-4, -5), (7, -2), (9, 1)} and {(-4, 1), (0, 3), (0, 0), (6, -7)} from above. One is a function and one is not. Which one is the function?

Look in each set to see if the

Below are input/output tables. Let's determine which table represents a function.

a)

b)

Think back to the definition of a function, “it cannot be a function if at least one input has more than one output.” a) has one input that has more than one output. Therefore, it is not a function. b) does not have an input with two different outputs. Therefore, it is a function.

Another way to approach these problems is to write out the points that are created and then determine if the

We can also apply this idea to equations. Every equation is a relation, but not every equation is a function. The easiest way to determine if an equation is a function is to do the **Vertical Line Test.** The Vertical Line Test will help you determine if any

Now, let's determine if the equation

The graph of

Drawing vertical lines through the circle, we see that it touches them twice.

This tells us that a circle is not a function.

However, if we solve the equation for

### Examples

#### Example 1

Earlier, you were asked if the vending machine is an example of a function.

Each input (a letter/number combination) into the vending machine results in one and only one item. Since none of the items appear more than once in the vending machine, it is an example of a function.

#### Example 2

Determine if the following relation is a function: {(3, -5), (8, 1), (-3, -3), (5, 1)}. Briefly explain your answer.

Yes, the

#### Example 3

Determine if the following relation is a function: {(9, -2), (0, 0), (7, 4), (9, 3)}. Briefly explain your answer.

No, there are two 9’s in the

#### Example 4

Determine if the relation below is a function. Briefly explain your answer.

Yes, all the outputs have different inputs.

#### Example 5

Determine if the relation below is a function. Briefly explain your answer.

Yes, this graph passes the Vertical Line Test. Any vertical line touches this graph once.

### Review

Determine if the following relations are functions. Briefly explain your answer.

- {(3, 4), (5, 6), (7, 8), (9, 10)}
- {(-9, -10), (4, -5), (6, -5), (4, -10)}
- {(-5, -7), (0, 4), (-5, 3), (9, 4)}
- {(3, -12), (6, -1), (-10, 5), (-2, 9)}

HINT: Recall that with open circles, the point is not included.

For problems 17-19, determine if the following lines are functions.

y=−3x−1 y=23x+6 y=−2 - Is
x=4 a function? Why or why not? - From problems 17-20, what can you conclude about linear equations?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 2.8.