Functions and Relations

Objective: To find values of, graph, and determine the domain and range of relations as well as determine which relations are functions.

by Gavin McGerald 2

Relations

• A relation is any set of ordered pairs. The ordered pairs are made up of a domain (typically x) which is mapped to a range (typically y or f(x)).

by Gavin McGerald 3

What is a mapping?

• Domain

-2

3

5

• Range

0

2

4

7

In this relation {(-2, 2), (3, 0), (3, 7), (5, 0), (5, 2),(5, 4)} the domain {-2, 3, 5} is mappedto the range {0, 2, 4, 7}

by Gavin McGerald 4

What is a function?

• A function is a relation where each element of the domain is paired with exactly one element of the range.

by Gavin McGerald 5

Relations that are:

Functions• {(2,3), (3,1), (1,1)}• 2 3

1 40

•

Not Functions• {(3,2), (1,3), (1,1)}• 2 3

1 40

• y

x

y

x

by Gavin McGerald 6

How can you tell if a graph of a relation is a function?

• The vertical line test.

• A relation is a function if and only if no vertical line intersects its graph more than once.

by Gavin McGerald 7

Terminology

• f: x 3x is read “the function f that assigns x to the number 3x

• Function notationf(x) “function of x” or “f of x”

by Gavin McGerald 8

Try These!

• g: x → 2x – 3

1. g(3)

2. g(½)

3. g(-3)

4. g(h + 1)

5. g(x + ½)

• Solutions• 2(3) – 3 = 3• 2(½) – 3 = -2• 2(-3) – 3 = -9• 2(h + 1) – 3 = 2h – 1 • 2(x + ½) – 3 = 2x – 2

by Gavin McGerald 9

Try A Few More!

• Determine if each relation is a function.

1. {(2, 3), (-1, 0), (1, 3), (0, 2)}

2. 3.

4. y = 3x – 2 5. y2 = x

y

x

y

x

by Gavin McGerald 10

Solutions

1. Yes

2. Yes

3. No, vertical line test

4. Yes, its an equation for a line and thus a vertical line will only pass through it once.

5. No, If x = 4 y could equal 2 or -2