# Relations and Functions

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Relations and Functions
1-6 Relations and Functions Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Warm Up Use the graph for Problems 1–2.
1. List the x-coordinates of the points. 2. List the y-coordinates of the points. –2, 0, 3, 5 3, 4, 1, 0

Objectives Identify the domain and range of relations and functions.
Determine whether a relation is a function.

Vocabulary relation domain range function

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 values for a relation is called the domain, and the set of output values is called the range.

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

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.

Give the domain and range for the relation shown in the graph.
Check It Out! Example 1 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.

Suppose you are told that a person entered a word into a text message using the numbers 6, 2, 8, and 4 on a cell phone. It would be difficult to determine the word without seeing it because each number can be used to enter three different letters.

Number {Number, Letter} {(6, M), (6, N), (6, O)}
The numbers 6, 2, 8, and 4 each appear as the first coordinate of three different ordered pairs. {(6, M), (6, N), (6, O)} {(2, A), (2, B), (2, C)} {(8, T), (8, U), (8, V)} {(4, G), (4, H), (4, I)}

However, if you are told to enter the word MATH into a text message, you can easily determine that you use the numbers 6, 2, 8, and 4, because each letter appears on only one numbered key. The first coordinate is different in each ordered pair. {(M, 6), (A, 2), (T, 8), (H,4)} A relation in which the first coordinate is never repeated is called a function. In a function, there is only one output for each input, so each element of the domain is mapped to exactly one element in the range.

Although a single input in a function cannot be mapped to more than one output, two or more different inputs can be mapped to the same output.

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.

Example 2: Determining Whether a Relation is a Function
Determine whether each relation is a function. A. from the items in a store to their prices on a certain date There is only one price for each different item on a certain date. The relation from items to price makes it a function. B. from types of fruits to their colors A fruit, such as an apple, from the domain would be associated with more than one color, such as red and green. The relation from types of fruits to their colors is not a function.

Check It Out! Example 2 Determine whether each relation is a function. A. There is only one price for each shoe size. The relation from shoe sizes to price makes is a function. B. from the number of items in a grocery cart to the total cost of the items in the cart The number items in a grocery cart would be associated with many different total costs of the items in the cart. The relation of the number of items in a grocery cart to the total cost of the items is not a function.

Every point on a vertical line has the same x-coordinate, so a vertical line cannot represent a function. If a vertical line passes through more than one point on the graph of a relation, the relation must have more than one point with the same x-coordinate. Therefore the relation is not a function.

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.

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 not a function. A vertical line at x = 1 would pass through (1, 1) and (1, –2).

Check It Out! Example 3a 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.

Check It Out! Example 3a 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).

Lesson Quiz: Part I 1. Give the domain and range for this relation: {(10, 5), (20, 5), (30, 5), (60, 100), (90, 100)}. Determine whether each relation is a function. 2. from each person in class to the number of pets he or she has 3. from city to zip code D: {10, 20, 30, 60, 90)} R: {5, 100} function not a function

Lesson Quiz: Part II Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. 4. not a function; possible answer: (3, 2) and (3, –2)

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