Chapter 1 Chapter 1 A Beginning Library of Elementary A Beginning Library of Elementary

FunctionsFunctions

Section 1Section 1

FunctionsFunctions

Definition of a functionDefinition of a function

• A Function is a rule (process or method) A Function is a rule (process or method) that produces a correspondence between that produces a correspondence between two sets of elements such that to each two sets of elements such that to each element in the first set there corresponds element in the first set there corresponds one and only one element in the second one and only one element in the second set.set.

• The first set is called the domain (x values) The first set is called the domain (x values) and the second set is called the range (y and the second set is called the range (y values).values).

ExamplesExamples

Function

Not a Function

Function

-2 -8

-1 -1

0 0

1 1

2 8

Domain

Range 0 0

11 -1

24 -2

39 -3

Domain

Range

-2 4

-1 1

0 0

1

2

Domain

Range

Vertical Line TestVertical Line Test

• An equation defines a function if An equation defines a function if each vertical line in the coordinate each vertical line in the coordinate system passes through at most one system passes through at most one point on the graph of the equation.point on the graph of the equation.

• If any vertical line passes through If any vertical line passes through two or more points on the graph of two or more points on the graph of an equation, then the equation does an equation, then the equation does not define a function.not define a function.

ExamplesExamples

Not a Function

Function

Not a Function

Functions Defined by Functions Defined by EquationsEquations

• If in an equation in two variables, we If in an equation in two variables, we get exactly one output (value for the get exactly one output (value for the dependent variable) for each input dependent variable) for each input (value for the independent variable), (value for the independent variable), then the equation defines a function.then the equation defines a function.

• If we get more than one output for a If we get more than one output for a given input, the equation does not given input, the equation does not define a function.define a function.

ExamplesExamples

4 3 8

4 3 8

3 8

4

y x

y x

xy

2 2

2 2

2 2

2

9

9

9

9

y x

y x

y x

y x

Example 1 Example 2

Function Not a

Function

Function NotationFunction Notation

• For any element x in the domain of the For any element x in the domain of the function f, the symbol f(x) represents function f, the symbol f(x) represents the element in the range of f the element in the range of f corresponding to x in the domain of f. corresponding to x in the domain of f. If x is an input value, then f(x) is the If x is an input value, then f(x) is the corresponding output value. If x is an corresponding output value. If x is an element that is not in the domain of f, element that is not in the domain of f, then f is not defined at x and f(x) does then f is not defined at x and f(x) does not exist.not exist.

Function EvaluationFunction Evaluation

Example 1 Example 2

2

22 2

( ) 1

( ) 1 ( )

1 4 3

g x x

g

12( )

212 12

( )2

6 346

f xx

f

Example 3

2 2

( ) 1

( ) 1

3

h x x

h

Not a Real Number

Domains and Ranges of a Domains and Ranges of a FunctionFunction

• If a function is specified by an equation If a function is specified by an equation and the domain is not indicated, then we and the domain is not indicated, then we assume that the domain is the set of all assume that the domain is the set of all real number replacements of the real number replacements of the independent variable (inputs) that independent variable (inputs) that produce real values for the dependent produce real values for the dependent variable (outputs). The range is the set variable (outputs). The range is the set of all outputs corresponding to input of all outputs corresponding to input values.values.

Finding the Domain of a Finding the Domain of a FunctionFunction

• ProblemsProblems– Zero in the denominatorZero in the denominator– Negative numbers under square rootsNegative numbers under square roots

• Take values of x that cause problems Take values of x that cause problems out of the domainout of the domain