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?

x

y

Example

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

x

y

Example

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

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?

x

y

Even and Odd Functionsand Symmetry

A graph is symmetric with respect to they-axis if, for every point (x,y) on the graph,the point (-x,y) is also on the graph. All evenfunctions 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?

x

y

Example

Is this an even or odd function?

x

y

Example

Is this an even or odd function?

x

y

Piecewise Functions

A function that is defined by two or more equations overa 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 6020 0.40( 60) if t>60t

Example

Find and interpret each of the following.

C t 20 if 0 t 6020 0.40( 60) if t>60t

45

60

90

C

C

C

ExampleGraph the following piecewise function.

f x 3 if - x 32 3 if x>3x

x

y

Functions and Difference Quotients

See next slide.

2

2

2 2

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

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 5h

f(x+h) from the previous slidef(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

hhx h x h x x

h

2 2

2 2

2x+h-2

hx h hh

h x hh

Example

Find and simplify the expressions iff(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 functionsbecause their graphs form discontinuous steps. One suchfunction is called the greatest integer function, symbolizedby int(x) or [x], whereint(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)=1int(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 letters2 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.

x

y

$1.00$ .75$ .50$ .25

(a)

(b)

(c)

(d)

There is a relative minimum at x=?

43

20

x

y

(a)

(b)

(c)

(d)

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

2

6

3 666

x xhx hx

(a)

(b)

(c)

(d)

Evaluate the following piecewise function at f(-1) 2x+1 if x<-1f(x)= -2 if -1 x 1 x-3 if x>1

24

01