Graphing Relations, Domain, and Range

**Graphing Relations, Domain, and Range**

A **relation** is just a relationship between sets of information. When *x* and *y* values are linked in an equation or inequality, they are related; hence, they represent a relation.

Not all relations are functions. A **function** states that given an *x*, we get one and only one *y*.

y = 3x + 1 | x^{2} + y^{2} = 5 |

This is a function. For any value of x you plug in, you will get only one possible value for y. For example, if x = 2, y can only equal 7. | This is not a function. Any value of x can give you more than one possible y. For example, if x = 1, y could equal 2 or −2. |

It is possible to test a graph to see if it represents a function by using the **vertical line test**. Given the graph of a relation, if you can draw a vertical line that crosses the graph in more than one place, then the relation is not a function.

This is a function. Any vertical line will cross this graph at only one point. | This is not a function. There is a vertical line that will cross this graph at more than one point. |

The **domain** is defined as all the possible input values (usually *x*) which allow the formula to work. Note that values that cause a denominator to be zero, which makes the function undefined, are not allowable values.

- The function
*y*= 4*x*^{2}− 9 has a domain of all real numbers, which can be expressed using the interval (−∞, ∞). Every possible*x*-value will give you a legitimate*y*-value.

- The function has a domain of all real numbers except −5, because when
*x*= −5, the denominator will be zero, and the function will be undefined. We can express this using the inequality*x*≠ −5, or the interval (−∞, −5) ⋃ (−5, ∞).

The **range** is the set of all possible output values (usually *y*), which result from using the formula.

*y*=

*x*

^{2}− 2

*x*− 1, you'll see that the

*y*-values begin at −2 and increase forever. The range of this function is all real numbers from −2 onward. We can express this using the interval [−2, ∞).

Practice

Practice

For each relation, state if it is a function. State the domain and range, and sketch a graph.

1.

2.

3.

4.

5.

**Answer Key**

1. Not a function

2. Function

3. Function

4. Function

5. Function