Definition of a Function

A function is a set of ordered pairs in which no two ordered pairs of the form (x, y) have the same x-value with different y-values.

That sounds easy enough. Maybe we should look at some

examples.

Example 1

(-3, 9) (-2, 4) (-1, 1) (0, 0) (1, 1) (2, 2) (3, 3)This is a function because there are no repeating x-values.

Example 2

(5, 7) (3, 5) (1, 4) (0, 0) (1, 6) (4, 2) (6, 3) This is not a function because there are repeating x-values.

Example 2 is a relation.

Representing a FunctionSuppose a plumber gets paid $40 for traveling time to a job plus $60 for each hour he takes to complete the job. The plumber charges for whole-number hours. The function that shows the relationship between the number of hours, x, that the plumber works and the amount of money, y, that he charges can be represented in different ways.

Algebraic Representation

40 60y x

Table Representation

1 100

2 160

3 220

Graphical Representation

0 1 2 3 4 5 6

440

400

360320280240200

160

120 80 40

(1, 100)

(2, 160)

(3, 220)

(4, 280)

(5, 340)

(6, 400)

x hours

y dollars

4 280

5 3406 400

Function NotationFor any function f, the notation f(x) is read “ f of x” and represents the value of y when x is replaced by the number or expression inside the parenthesis.

( )y f xInput ValueIndependent Variable

f(x)}

x}Output ValueDependent Variable

DomainSet of all independent variables for which a function is defined.

All x-values.

RangeSet of all dependent variables for which a function is defined.

All values of f(x).

Domain

Range

Evaluating a Function

2( ) 2xf x

Evaluate f(3)

2() 23)3(f

2(3) 9f

1(3) 1f

22) 7 3(x x xf

Evaluate f(-2)

2( ) (2 3) 722 2)(f

(2( ) 4)2 14 3f

8) 32 14(f

9( 2)f

( 5) 2xf x

Evaluate f(x+3)

2 )3 5(( ) 3x xf

2) 6( 53f x x

( ) 13 2f xx

This is really easy. I better push the easy

button.

Determining the Domain and Range of a Function

What are the domain and range of the following functions?

That means find all the x-values for which the function is defined, and all the

corresponding y-values.

( ) 2f x x Since the value under the radical sign can’t be negative, the x value can’t be less than 2.Domain { }2 x

With this domain, the value of the function will never be less than 0.

Ra ge }n { 0 y

2

39

( )x

xfx

Since the denominator can’t be zero, the x value can’t be positive 3 or negative 3.Domain 3 { }x

With this domain, the value of the function can be any real number.

Range all real numbe{ rs}

Vertical Line Test

A graph represents a function if any vertical line drawn intersects it in at most one point.

The vertical line never intersects the graph in more than one point, therefore this is a function.

The vertical line never intersects the graph in more than one point, therefore this is a function.

The vertical line does intersect the graph in more than one point, therefore this is not a function.

Graphs of Relations and Functions

2y x

2x y2y

2x y xx y

Is a Functio

n Not a Function

Is a Functio

n

Not a Function

Is a Functio

n

Not a Function

Equations of FunctionsLet’s take a look at the equations from the previous page.

Equations that are Functions

2y x

2y

y x

Equations thatare not Functions

2x y

2x

x y

y cannot be squared

x cannot be a constant

y cannot be absolute value

That was easy

One-to-One Functions

A function is a One-to-One Function if the same value of y is never associated with two different values of x.

That means a One-to-One Function cannot have repeating x values or

repeating y values.

(-3, -9) (-2, -7) (-1, -5) (0, -3) (1, -1) (2, 1) (3, 3)

This is a One-to-One Function because there are no repeating y values.

(-3, 9) (-2, 4) (-1, 1) (0, 0) (1, 1) (2, 4) (3, 9)

This is not a One-to-One Function because there are repeating y values.

Horizontal Line TestA graph represents a one-to-one function if any horizontal line drawn intersects it in at most one point.

The horizontal line never intersects the graph in more than one point, therefore this is a one-to-one function.

The horizontal line does intersect the graph in more than one point, therefore this is not a one-to-one function.

The horizontal line does intersect the graph in more than one point, therefore this is not a one-to-one function.