Transcript 1.3
Section 1.5
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.
Find where the graph is increasing?
Where is it decreasing? Where is it
constant?
Example
y
x
Example
y
Find where the graph is increasing? Where
is it decreasing? Where is it constant?
x
Example
Find where the graph is increasing? Where
is it decreasing? Where is it constant?
y
x
Relative Maxima
And
Relative Minima
Where are the relative minimums?
Where are the relative maximums?
Example
Why are the maximums and minimums
called relative or local?
y
x
Even and Odd Functions
and 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 thirdquadrant 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?
y
x
Example
Is this an even or odd function?
y
x
Example
Is this an even or odd function?
y
x
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(t 60) if t>60
Example
C t
20
if 0 t 60
20 0.40(t 60) if t>60
Find and interpret each of the following.
C 45
C 60
C 90
Example
Graph the following piecewise function.
3
f x
if - x 3
y
2 x 3 if x>3
x
Functions and
Difference Quotients
See next slide.
f(x+h)-f(x)
2
Find
for f(x)=x 2 x 5
h
First find f(x+h)
f(x+h)=(x+h) 2(x+h)-5
2
x 2hx h 2 x 2h 5
2
2
Continued on the next slide.
f(x+h)-f(x)
Find
for f(x)=x 2 2 x 5
h
Use f(x+h) from the previous slide
f(x+h)-f(x)
Second find
h
2
2
2
x
2
hx
h
2
x
2
h
5
x
2 x 5
f(x+h)-f(x)
h
h
x 2 2hx h 2 2 x 2h 5 x 2 2 x 5
h
2hx h 2 2h
h
h 2x h 2
h
2x+h-2
Example
Find and simplify the expressions if f ( x) 2 x 1
Find f(x+h)
f(x+h)-f(x)
Find
, h0
h
Example
Find and simplify the expressions if
Find f(x+h)
f ( x) x 4
2
f(x+h)-f(x)
Find
, h0
h
Example
2
f
(
x
)
x
2x 1
Find and simplify the expressions if
Find f(x+h)
f(x+h)-f(x)
Find
, h0
h
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.
y
b. The charge for a letter that weights 2.3 oz.
$1.00
$ .75
$ .50
$ .25
x
y
There is a relative minimum at x=?
(a) 4
(b) 3
(c) 2
(d) 0
Find the difference quotient for f(x)=3x 2 .
(a) 6
(b) 3 x 2 6 xh
(c) 6 x h
(d) 6 x
Evaluate the following piecewise function at f(-1)
2x+1 if x<-1
f(x)=
(a) 2
(b) 4
(c) 0
(d) 1
-2 if -1 x 1
x-3 if x>1