2.3 Properties of Functions 2008
October 14, 2008
2.3 Properties of Functions Objectives: • Determine even and odd functions.
• Locate and identify local maxima and minima.
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2.3 Properties of Functions 2008
October 14, 2008
A. Determine Even and Odd Functions
1. A function is even if, for every (x, y) on the graph of f(x), there is also a (­x, y).
(­2, 4)
(2, 4)
Oct 10­10:58 AM
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2.3 Properties of Functions 2008
October 14, 2008
2. Testing to see if a function is even
g(x) = x2 ­ 5
a. Graphically: Use a graphing calculator to see if it has y­axis symmetry.
b. Algebraically: f(­x) = f(x)
g(x) = x2 ­ 5
g(­x) = (­x)2 ­ 5 = x2 ­ 5 = g(x)
c. Numerically: Test a value and its opposite.
Let x = 2
g(2) = (2)2 ­ 5 = ­1
Let x = ­2
g(­2) = (­2)2 ­ 5 = ­1
Oct 10­3:38 PM
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2.3 Properties of Functions 2008
October 14, 2008
3. A function is odd if, for every (x, y) on the graph of f(x), there is also a (­x, ­y).
(2, 8)
(1, 4)
4
2
­2
­2
­4
1
2
(­1, ­ 4)
(­2, ­8)
Oct 10­11:01 AM
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2.3 Properties of Functions 2008
October 14, 2008
4. Testing to see if a function is odd
h(x) = 5x3 ­ x
a. Graphically: Use a graphing calculator to see if it has symmetry
with respect to the origin.
b. Algebraically: f(­x) = ­f(x)
h(x) = 5x3 ­ x
h(­x) = 5(­x)3 ­ (­x) = ­5x3 + x = ­(5x3 ­ x)
c. Numerically: Test a value and its opposite.
Let x = 2
Let x = ­2
h(2) = 5(2)3 ­ 2
h(­2) = 5(­2)3 ­ (­2)
= 5(8) ­ 2
= 5(­8) + 2
= 40 ­ 2
= ­40 + 2
= 38
= ­38
Oct 10­3:46 PM
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2.3 Properties of Functions 2008
October 14, 2008
B. Increasing, Decreasing, and Constant
Oct 10­11:02 AM
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2.3 Properties of Functions 2008
October 14, 2008
Interval Notation
­5
­4
­3
­2
­1
­5 < x < 2
­5
­4
­3
­2
­1
­5 < x < 2
0
1
is
0
2
3
4
5
[­5, 2]
1
is
2
3
4
5
(­5, 2)
Oct 4 ­ 8:29 AM
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2.3 Properties of Functions 2008
October 14, 2008
Interval Notation
­5
­4
­3
­2
­1
­5 < x < 2
­5
­4
­3
­2
­1
­5 < x < 2
0
1
is
0
2
3
4
5
4
5
[­5, 2)
1
is
2
3
(­5, 2]
Oct 4 ­ 8:32 AM
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2.3 Properties of Functions 2008
October 14, 2008
1. Local Maxima: the y­value at a high point
2. Local Minima: the y­value at a low point
3. Example:
(1, 2)
(­1, 1)
(3, 0)
a. At what number(s) does f have a local maximum?
b. What is the local maximum?
c. At what number(s) does f have a local minimum?
d. What are the local minima?
e. List the intervals on which f is increasing.
f. List the intervals on which f is decreasing.
Oct 10­11:07 AM
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2.3 Properties of Functions 2008
October 14, 2008
Homework: page 89 (11 ­ 20, 22 ­ 28 even, 29 ­ 32, 34 ­ 44 even, 63)
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