Sullivan College Algebra: Section 3.3 Properties of Functions

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Sullivan College Algebra Section 3.3Properties
of Functions

• Objectives
• Determine Even and Odd Functions from a Graph
• Identify Even and Odd Functions from the
  Equation
• Determine Where a Function is Increasing,
  Decreasing, or is Constant
• Locate Maxima and Minima
• Find the Average Rate of Change of a Function

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A function f is even if for every number x in its
domain the number -x is also in the domain and
f(x) f(-x).
A function is even if and only if its graph is
symmetric with respect to the y-axis.
A function f is odd if for every number x in its
domain the number -x is also in the domain and
-f(x) f(-x).
A function is odd if and only if its graph is
symmetric with respect to the origin.
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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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Determine whether each of the following functions
is even, odd, or neither. Then determine whether
the graph is symmetric with respect to the y-axis
or with respect to the origin.
a.)
Even function, graph symmetric with respect to
the y-axis.
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b.)
Not an even function.
Odd function, and the graph is symmetric with
respect to the origin.
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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
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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.