Section 2.2 More on Functions and Their Graphs

Increasing and Decreasing Functions

The open intervals describing where functions increase, decrease, or are constant, use x-coordinates and not the y-coordinates.

Example Find where the graph is increasing? Where is it decreasing? Where is it constant?

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

x

y

Example

Find where the graph is increasing? Where is it decreasing? Where is it constant?

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

x

y

Example

Find where the graph is increasing? Where is it decreasing? Where is it constant?

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

x

y

Relative Maxima And

Relative Minima

Example Where are the relative minimums? Where are the relative maximums?

Why are the maximums and minimums called relative or local?

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

x

y

Even and Odd Functionsand Symmetry

A graph is symmetric with respect to the

y-axis if, for every point (x,y) on the graph,

the point (-x,y) is also on the graph. All even

functions have graphs with this kind of symmetry.

A graph is symmetric with respect to the origin if,

for every point (x,y) on the graph, the point (-x,-y)

is also on the graph. Observe that the first- and third-

quadrant portions of odd functions are reflections of

one another with respect to the origin. Notice that f(x)

and f(-x) have opposite signs, so that f(-x)=-f(x). All

odd functions have graphs with origin symmetry.

Example

Is this an even or odd function?

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

x

y

Example

Is this an even or odd function?

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

x

y

Example

Is this an even or odd function?

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

x

y

Piecewise Functions

A function that is defined by two or more equations over

a specified domain is called a piecewise function. Many

cellular phone plans can be represented with piecewise

functions. See the piecewise function below:

A cellular phone company offers the following plan:

$20 per month buys 60 minutes

Additional time costs $0.40 per minute.

( )

C t =20 if 0 t 60

20 0.40( 60) if t>60t

≤ ≤+ −

Example

Find and interpret each of the following.

( )

C t =20 if 0 t 60

20 0.40( 60) if t>60t

≤ ≤+ −

( )( )( )

45

60

90

C

C

C

Example

Graph the following piecewise function.

( )

f x =3 if - x 3

2 3 if x>3x

∞ ≤ ≤−

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

x

y

Functions and Difference Quotients

See next slide.

2

2

2 2

f(x+h)-f(x) for f(x)=x 2 5

h f(x+h)

f(x+h)=(x+h) 2(x+h)-5

x 2 2 2 5

Find x

First find

hx h x h

− −

−+ + − − −

Continued on the next slide.

( )

2

2 2 2

2 2 2

f(x+h)-f(x) for f(x)=x 2 5

h f(x+h) from the previous slide

f(x+h)-f(x) find

h

x 2 2 2 5 x 2 5f(x+h)-f(x)

h

x 2 2 2 5 2 5

2

Find x

Use

Second

hx h x h x

h

hx h x h x x

h

− −

+ + − − − − − −=

+ + − − − − + +

( )

2 2

2 2

2x+h-2

hx h h

hh x h

h

+ −

+ −

Example

Find and simplify the expressions if

f(x+h)-f(x)Find f(x+h) Find , h 0

h≠

( ) 2 1f x x= +

Example

Find and simplify the expressions if

f(x+h)-f(x)Find f(x+h) Find , h 0

h≠

2( ) 4f x x= −

Example

Find and simplify the expressions if

f(x+h)-f(x)Find f(x+h) Find , h 0

h≠

2( ) 2 1f x x x= − +

Some piecewise functions are called step functions

because their graphs form discontinuous steps. One such

function is called the greatest integer function, symbolized

by int(x) or [x], where

int(x)= the greatest integer that is less than or equal to x.

For example,

int(1)=1, int(1.3)=1, int(1.5)=1, int(1.9)=1

int(2)=2, int(2.3)=2, int(2.5)=2, int(2.9)=2

Example

The USPS charges $ .42 for letters 1 oz. or less. For letters

2 oz. or less they charge $ .59, and 3 oz. or less, they charge $ . 76.

Graph this function and then find the following charges.

a. The charge for a letter that weights 1.5 oz.

b. The charge for a letter that weights 2.3 oz.

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

−7

−6

−5

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−2

−1

1

2

3

4

5

6

x

y

$1.00

$ .75

$ .50

$ .25

(a)

(b)

(c)

(d)

There is a relative minimum at x=?

4

3

2

0

−

−−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10

−7

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

x

y

(a)

(b)

(c)

(d)

2Find the difference quotient for f(x)=3x .

2

6

3 6

6

6

x xh

x h

x

++

(a)

(b)

(c)

(d)

Evaluate the following piecewise function at f(-1)

2x+1 if x<-1

f(x)= -2 if -1 x 1

x-3 if x>1

≤ ≤

2

4

0

1

−−

−