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