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**Objectives Identify the domain and range of relations and functions.**

Determine whether a relation is a function. Vocabulary Words Relation and Function Vertical Line Test Domain and Range

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**A relation is a pairing of input values with output values**

A relation is a pairing of input values with output values. It can be shown as a set of ordered pairs (x,y), where x is an input and y is an output. The set of input (x) values for a relation is called the domain, and the set of output (y) values is called the range.

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Mapping Diagram Domain Range A 2 B C Set of Ordered Pairs {(2, A), (2, B), (2, C)} (x, y) (domain, range)

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**Example 1: Identifying Domain and Range**

Give the domain and range for this relation: {(100,5), (120,5), (140,6), (160,6), (180,12)}. List the set of ordered pairs: {(100, 5), (120, 5), (140, 6), (160, 6), (180, 12)} Domain: {100, 120, 140, 160, 180} The set of x-coordinates. Range: {5, 6, 12} The set of y-coordinates.

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**Give the domain and range for the relation shown in the graph.**

Example 2 Give the domain and range for the relation shown in the graph. List the set of ordered pairs: {(–2, 2), (–1, 1), (0, 0), (1, –1), (2, –2), (3, –3)} Domain: {–2, –1, 0, 1, 2, 3} The set of x-coordinates. Range: {–3, –2, –1, 0, 1, 2} The set of y-coordinates.

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Not a function: The relationship from number to letter is not a function because the domain value 2 is mapped to the range values A, B, and C. Function: The relationship from letter to number is a function because each letter in the domain is mapped to only one number in the range. Both functions and non-functions are relations!

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**Example 3A: Using the Vertical-Line Test**

Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. This is a function. Any vertical line would pass through only one point on the graph.

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**Example 3B Using the Vertical-Line Test**

Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. This is a function. Any vertical line would pass through only one point on the graph.

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**Example 3C: Using the Vertical-Line Test**

Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. This is not a function. A vertical line at x = 1 would pass through (1, 1) and (1, –2).

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**Example 4A: Is this relation a function?**

Function: The relationship is a function if each member of the domain is mapped to only one number in the range. Example 4A: Is this relation a function? From graph or ordered pairs: {(–2, 2), (–1, 1), (0, 0), (1, –1), (2, –2), (3, –3)}

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**Example 4B: Is this relation a function?**

Function: The relationship is a function if each member of the domain is mapped to only one number in the range. Example 4B: Is this relation a function? {(100,5), (120,5), (140,6), (160,6), (180,12)}. Identify the domain and range: Domain: {100, 120, 140, 160, 180} Range: {5, 6, 12}

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**Example 4C: Is this relation a function?**

There is only one price for each shoe size. The relation from shoe sizes to price makes is a function. B. Names in a classroom and grades C. (Reverse B…) from class grades, find student name.

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**Example 5A: Is this relation a function?**

Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. This is not a function. A vertical line at x = 1 would pass through (1, 2) and (1, –2).

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**Example 5B: Is this relation a function?**

Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. not a function

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