2.1 Relations and Functions

In this chapter, you will learn:

What a function is. Review domain and range. Linear equations. Slope. Slope intercept form y = mx+b. Point-slope form y – y1 = m(x – x1). Linear regression.

What is a function?

A function is a special type of relation in which each type of domain (x values) is paired of with exactly one range value (y value).

FUNCTION FUNCTION

NOT A FUNCTION

FUNCTIONFUNCTION

NOT A FUNCTION

NOT A FUNCTION

Relations and Functions

-3

0

2

1

2

4

Suppose we have the relation { (-3,1) , (0,2) , (2,4) }

FUNCTIONONE – TO – ONE

DOMAIN x - values

RANGE y - values

Relations and FunctionsSuppose we have the relation { (-1,5) , (1,3) , (4,5) }

-1

1

4

5

3

5

FUNCTIONNOT ONE – TO – ONE

Relations and FunctionsSuppose we have the relation { (5,6) , (-3,0) , (1,1) , (-3,6) }

5

-3

1

0

1

6

NOT A FUNCTION

Domain and Range

The set of all inputs, or x-values of a function.It is all the x – values that are allowed to be used.

The set of all outputs, or y-values of a function.It is all the y – values that are represented.

Example 1

Domain = ________________ Range = _________________

All x – values or (-∞ , ∞)

Just 4 or {4}

Example 2

Domain = ________________ Range = _________________All y – values or (-∞ , ∞)

Just -5 or {-5}

Example 3

Domain = ________________ Range = _________________

All x – values or (-∞ , ∞)

From -6 on up or [-6 , ∞)

Example 4

Domain = ________________ Range = _________________All y – values or (-∞ , ∞)

From -6 on up or [-6 , ∞)

Example 5

Domain = ________________ Range = _________________All y – values or (-∞ , ∞)

All x – values or (-∞ , ∞)

Function Notation

Function notation, f(x) , is called “f of x” or “a function of x”.

It is not f times x .

Example: if y = x+2 then we say f(x) = x+2. If y = 5 when x = 3, then we say f(3) = 5

What is function notation?

Example 1f(x) = 3x + 1

f( 13) = ____________________

f( 5) = ____________________

f( -11) = ____________________

3 (5) + 1 = 16

3 (13) + 1 = 40

3 (-11) + 1 = -32

Example 2f(x) = x² + 3x - 5

f( 0) = ____________________

f( 5) = ____________________

f( 4) = ____________________

5² + 3 (5) – 5 = 35

0² + 3 (0) – 5 = -5

4² + 3 (4) – 5 = 23