FunctionsFunctions P.5P.5

ObjectivesDetermine whether relations between two

variables are functions.Use function notation and evaluate functions.Find the domain of a function.Use functions to model and solve real life

problems.Evaluate difference qutients.

Two quantities that are related to one another by some rule are called a relation.

A function consists of:

A set of inputs (the domain or x values) - independent

A rule by which each input determines exactly one output

A set of outputs (the range or y values) - dependent

Example

Solution The domain is the set of all first components. Thus, the domain is

{1994,1995,1996,1997,1998}.

The range is the set of all second components. Thus, the range is

{56.21, 51.00, 47.70, 42.78, 39.43}.

Find the domain and range of {(1994, 56.21), (1995, 51.00), (1996, 47.70),

(1997, 42.78), (1998, 39.43)}

Example

Solution We begin by making a figure for each relation that shows set X, the domain, and set Y, the range, shown below.

Determine whether the following are functions.

a. {(1, 6), (2, 6), (3, 8), (4, 9)} b. {(6,1),(6,2),(8,3),(9,4)}

1234

689

Domain Range

(a) Figure (a) shows that every element in the domain corresponds to exactly one element in the range. No two ordered pairs in the given relation have the same first component and different second components. Thus, the relation is a function.

689

1234

Domain Range

(b) Figure (b) shows that 6 corresponds to both 1 and 2. This relation is not a function; two ordered pairs have the same first component and different second components.

Characteristics of a Function from Set A to Set B

1. Each element in A must be matched with an element in B.

2. Some elements in B may not be matched with an element in A.

3. Two or more elements in A may be matched with the same element in B.

4. An element in A cannot be matched with two different elements in B.

Function Representation

Verbally – a sentence Numerically – a table or list of

ordered pairs Graphically – points on a graph

(input,output) Algebraically – equation in two

variables

EXAMPLES

Is y a function of x? Pg 68

Try it: Page 67 – 68 #12, 16, 22, 31, 36

f(x) denotes the output produced by the

input x also called the value of f at x

inputs are the domain or x values

outputs are the range or f(x) values

Function Notation

Example

If f (x) = x2 + 3x + 5, evaluate: a. f (2) b. f (x + 3)

If g (x) = , evaluate: a. g (2+t)

725 xx

Piecewise Functions Graphed in pieces. Each piece has a different domain.

2 1

2 1)(

2 xx

xxxf

Find: a. f(0) b. f(2) c. f(-1)

Finding Domains

Find the domain of the functions.

7)( 2 xxf

9

2)(

xxg

2rA

32)( xxg 3

3)(

x

xxf

Definition of a Difference Quotient The expression

for h≠0 is called the difference quotient.

Find:a. f(x) = (x+3) b. f(x) = (x2+1) c. f(x) = (1/x)

f (x h) f (x)

h

Groups Pg. 68 – 72 #42, 46, 48, 58, 60, 62, 66,

68, 80, 88, 98, 104,115 – 120.

Homework Pg. 67 – 72 # 4, 6, 7 – 35 0dd, 39, 41, 43,

45, 49, 51, 55, 57, 61, 65, 67, 71 – 81 odd, 87, 95, 103 – 109 odd