Section 2.2The Graph of a Function

The graph of f(x) is given below.

4

0

-4(0, -3)

(2, 3)

(4, 0) (10, 0)

(1, 0) x

y

What is the domain and range of f ?

Find f(0), f(4), and f(12)

Does the graph represent a function?

For which x is f(x)=0?

For what numbers is f(x) < 0?

For what value of x does f(x) = 3?

What are the intercepts (zeros)?

How often does the line x = 1 intersect the graph?

Section 2.3Properties of Functions

Example of an Even Function. It is symmetric about the y-axis x

y

(0,0)

x

y

(0,0)

Example of an Odd Function. It is symmetric about the origin

A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2).

A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) > f(x2).

A function f is constant on an open interval I if, for any choice of x in I, the values of f(x) are equal.

Determine where the following graph is increasing, decreasing and constant.

4

0

-4

(0, -3)

(2, 3)

(4, 0)

(10, -3)

(1, 0) x

y

(7, -3)

Increasing on (0,2)

Decreasing on (2,7)

Constant on (7,10)

A function f has a local maximum at c if there is an interval I containing c so that, for all x in I, f(x) < f(c). We call f(c) a local maximum of f.

A function f has a local minimum at c if there is an interval I containing c so that, for all x in I, f(x) > f(c). We call f(c) a local minimum of f.

Referring to the previous example, find all local maximums and minimums of the function: (hint: must be over an interval)

4

0

-4(0, -3)

(2, 3)

(4, 0)

(10, -3)

(1, 0) x

y

(7, -3)

Extreme Value Theorem: If a function f whose domain is a closed interval [a, b], then f has an absolute maximum and an absolute minimum on [a, b].

Note: An absolute max/min may also be a local max/min.

Referring to the previous example, find the absolute maximums and minimums of the function:

4

0

-4(0, -3)

(2, 3)

(4, 0)

(10, -3)

(1, 0) x

y

(7, -3)

If c is in the domain of a function y = f(x), the average rate of change of f between c and x is defined as

average rate of change =

yx

f x f cx c

( ) ( )

This expression is also called the difference quotient of f at c.

x - c

f(x) - f(c)

(x, f(x))

(c, f(c))

Secant Line

y = f(x)

The average rate of change of a function can be thought of as the average “slope” of the function, the change is y (rise) over the change in x (run).

Example: The function gives the height (in feet) of a ball thrown straight up as a function of time, t (in seconds).

610016)( 2 ttts

a. Find the average rate of change of the height of the ball between 1 and t seconds.

st

s t st

t

( ) ( )

,1

11

s t t t( ) 16 100 62

s( ) ( ) ( )1 16 1 100 1 6 902

s t st

t tt

( ) ( )

11

16 100 6 901

2

16 100 84

1

2t tt

4 4 25 21

1

2t t

t

4 4 25 21

1

2t t

t

4 4 21 1

1( )( )t t

t

4 4 21( )t

b. Using the result found in part a, find the average rate of change of the height of the ball between 1 and 2 seconds.

If t = 2, the average rate of change between 1 second and 2 seconds is: -4(4(2) - 21) = 52 ft/second.

Average Rate of Change between 1 second and t seconds is: -4(4t - 21)

Section 2.4Library of Functions;

Piecewise-Defined Functions

When functions are defined by more than one equation, they are called piecewise- defined functions.

Example: The function f is defined as:

1 3

1 3

12- 3

)(

xx

x

xx

xf

a.) Find f (1) = 3Find f (-1) = (-1) + 3 = 2

Find f (4) = - (4) + 3 = -1

1 3

1 3

12- 3

)(

xx

x

xx

xf

b.) Determine the domain of f

Domain: in interval notation),2[

or in set builder notation}2|{ xx

c.) Graph f

x

y

1 2 3

32

1

d.) Find the range of f from the graph found in part c.

Range: in interval notation)4,(

or in set builder notation}4|{ yy

x

y

1 2 3

32

1