Objectives q Identify the domain and range of

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

Vocabulary relation domain range function

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)

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

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. {(M, 6), (A, 2), (T, 8), (H, 4)} In each ordered pair, the first coordinate is different. The “x-coordinate” never repeats. 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.

A relation in which each member of the domain corresponds to exactly one member of the range is a function. Notice that more than one element in the domain can correspond to the same element in the range. Aerobics and tennis both burn 505 calories per hour. Is this relation a function? Yes

Determine whether each relation is a function? {(1, 8), (2, 9), (3, 10)} Yes {(3, 4), (5, 6), (3, 7)} No {(2, 6), (3, 6), (4, 6)} Yes

Functions as Equations

Not every set of ordered pairs defines a function. Not all equations with the variables x and y define a function. If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not define y as a function of x. So the equation is not a function. Determine whether each equation defines y as a function of x. Hint: Solve for y. Yes No

Function Notation

Evaluate each of the following.

The Vertical Line Test of a Function If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

Use the vertical line test to identify graphs in which y is a function of x. Not a function A function

Not a function A function

Yes

Obtaining Information From Graphs

Identifying Domain and Range from a Function’s Graph Using Interval Notation Find the coordinates of the endpoints. We write interval notation from least to greatest

Find the coordinates of the endpoints. We write interval notation from least to greatest.

Find the coordinates of the endpoints. We write interval notation from least to greatest.

Identifying Intercepts from a Function’s Graph