Section 2.2 The Graph of a Function

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Section 2.2The Graph of a Function
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The graph of f(x) is given below.
y
(2, 3)
4
(10, 0)
(4, 0)
0
x
(1, 0)
(0, -3)
-4
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Does the graph represent a function?
Find f(0), f(4), and f(12)
What is the domain and range of f ?
For which x is f(x)0?
What are the intercepts (zeros)?
For what value of x does f(x) 3?
For what numbers is f(x) lt 0?
How often does the line x 1 intersect the graph?
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Section 2.3Properties of Functions
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Example of an Even Function. It is symmetric
about the y-axis
Example of an Odd Function. It is symmetric
about the origin
(0,0)
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A function f is increasing on an open interval I
if, for any choice of x1 and x2 in I, with x1 lt
x2, we have f(x1) lt f(x2).
A function f is decreasing on an open interval I
if, for any choice of x1 and x2 in I, with x1 lt
x2, we have f(x1) gt 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.
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Determine where the following graph is
increasing, decreasing and constant.
Increasing on (0,2)
Decreasing on (2,7)
Constant on (7,10)
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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) lt 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) gt f(c). We call f(c) a local minimum of
f.
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Referring to the previous example, find all local
maximums and minimums of the function (hint
must be over an interval)
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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.
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Referring to the previous example, find the
absolute maximums and minimums of the function
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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
This expression is also called the difference
quotient of f at c.
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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).
y f(x)
Secant Line
(x, f(x))
f(x) - f(c)
(c, f(c))
x - c
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Example The function
gives the height (in feet) of a ball
thrown straight up as a function of time, t (in
seconds).
a. Find the average rate of change of the height
of the ball between 1 and t seconds.
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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.
Average Rate of Change between 1 second and t
seconds is -4(4t - 21)
If t 2, the average rate of change between 1
second and 2 seconds is -4(4(2) - 21) 52
ft/second.
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Section 2.4Library of FunctionsPiecewise-Define
d Functions
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When functions are defined by more than one
equation, they are called piecewise- defined
functions.
Example The function f is defined as
a.) Find f (1)
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Find f (-1)
(-1) 3 2
Find f (4)
- (4) 3 -1
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b.) Determine the domain of f
Domain in interval notation
or in set builder notation
c.) Graph f
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d.) Find the range of f from the graph found in
part c.
Range in interval notation
or in set builder notation