*Basics of Functions and Their Graphs

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. 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

B

C

2

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}

Range: {–3, –2, –1, 0, 1, 2}

The set of x-coordinates.

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 asthe first coordinate of three different orderedpairs.

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)}

{(3, 4), (5, 6), (3, 7)}

{(2, 6), (3, 6), (4, 6)}

Yes

No

Yes

Functions as Equations

Here is an equation that models paid vacation days eachyear as a function of years working for the company.

The variable x represents years working for a company. The variable y represents the average number of vacation dayseach year. The variable y is a function of the variable x. For each value of x, there is one and only one value of y. The variable x is called the independent variable because it can be assigned any value from the domain. Thus, x can be assigned any positive integer representing the number of years working for a company. The variable y is called the dependent variable because its value depends on x. Paid vacation days depend on years working for a company.

Not every set of ordered pairs defines a function.Not all equations with the variables x and y definea function. If an equation is solved for y and morethan 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 yas a function of x. Hint: Solve for y.

Yes

Yes

No

Function Notation

The special notation , read as represents the value of the function at the number, . If a function named , and represents the independent variable, the notation corresponds to the y-value for a given . The special notation is a fancy way of expressing the dependent variable, .

or

Evaluate (Hint: substitute 10 for every x variable.)

Evaluate each of the following.

Find f(2) for

Find f(−2) for

Find f(x + 2) for

𝟏𝟐

𝟓

𝒙𝟐+𝟐𝒙+𝟒

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

𝟎−𝟑

𝟐

Obtaining Information From Graphs

Identifying Domain and Range from a Function’s Graph

Using Interval Notation

Find the coordinates of the endpoints.

(−𝟏 ,𝟑)(𝟒 ,𝟏)

(𝟑 ,𝟎)

∞

We write interval notation from least to greatest

Find the coordinates of the endpoints.

Domain: Range:

(𝟑 ,𝟎)

−∞

We write interval notation from least to greatest.

∞

Find the coordinates of the endpoints.

(−𝟒 ,𝟐)

(𝟑 ,𝟎)

Domain: Range:

(−𝟐 ,−𝟐)

We write interval notation from least to greatest.

Domain: Range:

Identifying Intercepts from a Function’s Graph

𝑥𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑖𝑠−3 𝑓 (−4 )=2

𝑦 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡=2 𝑓 (2 )=3

𝑥𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠=0 𝑎𝑛𝑑4𝑦 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡=0

𝑓 (5 )=4