Even and Odd Functions
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Printed: December 3, 2014
AUTHOR
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C HAPTER
Chapter 1. Even and Odd Functions
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Even and Odd Functions
Objectives
In this concept, you learn how to determine if a function is even, odd, or neither.
Concept
While looking at the equation of a function that needed to be graphed for homework, a student recognized that the
task would go quickly because the sign of the values in the domain did not matter in calculating the function values,
and only half the values in the domain needed to be used. What kind of function was the student evaluating?
Watch This
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/57941
http://www.youtube.com/watch?v=8VgmBe3ulb8 - Khanacademy: Recognizing Odd and Even Functions
Guidance
A function f (x) is an even function if: f (x) = f (−x), i.e., the value of the function acting on an argument x is the
same as the value of the function when acting on the argument −x. So, for example, if f (x) is an even function,
then f (2) has the same answer as f (−2); f (5) has the same answer as f (−5), and so on.
Even functions are symmetric about the y-axis.
Example A
Show that the function y = f (x) = x2 is an even function.
Solution:
A function is an even function if f (x) = f (−x). For the above function, f (−x) = (−x)2 = x2 = f (x). So the function
is an even function. The function graph below shows that the function is symmetric with respect to the y-axis.
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In contrast to an even function, a function f (x) is an odd function if: − f (x) = f (−x), i.e., the function is odd
when the negative of the function’s answer for a given argument is the same as the function acting on the negative
argument. If a function f (x) were odd, then f (−2) = − f (2); f (−5) = − f (5), and so on.
Odd functions are not symmetric about the y-axis, but they are symmetric about the origin.
Example B
Show that the function y = x3 is odd.
Solution:
A function is an odd function if f (−x) = − f (x). For the above function, f (−x) = (−x)3 = −x3 = − f (x). So the
function is an odd function.
It is important to remember that a function does not have to be even or odd. Most functions are neither even nor
odd.
Example C
Determine whether the function y = 3(x + 2)2 + 4 is even or odd.
Solution:
Apply the test for an even function: f (−x) = 3(−x + 2)2 + 4 = 3(x − 2)2 + 4 6= f (x). The function is not an even
function.
Apply the test for an odd function: f (−x) = 3(x − 2)2 + 4 6= − f (x). The function is not an odd function.
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Chapter 1. Even and Odd Functions
The above function is neither even nor odd. As can be seen in the graph below, the function is neither symmetric
about the y-axis nor symmetric about the origin.
Concept question wrap-up
To be able to graph a function without having to worry about the sign of values in the domain means that the function
must be an even function, f (x) = f (−x), and is symmetric about the y-axis. Only half as many function values of the
dependent variable would be required to complete the graph. Either all negative values or all positive values could
be used to graph the function.
Vocabulary
An even function is a function that has the property: f (x) = f (−x); it is symmetric about the y-axis.
An odd function is a function that has the property: f (−x) = − f (x); it is symmetric about the origin.
Guided Practice
Here are eight basic functions that are often encountered. Use their function graphs to determine whether they are
even, odd, or neither.
1. Linear : f (x) = x
Domain = All reals
Range = All reals
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This linear function is symmetric about the origin and is an odd function: f (x) = f (−x).
2. Square(Quadratic) f (x) = x2
Domain = All reals
Range = {y ≥ 0}
As shown earlier in the concept, this quadratic function is symmetric about the y-axis and is an even function:
f (x) = f (−x).
3. Cube(Polynomial) f (x) = x3
Domain = All reals
Range = All reals
As shown earlier in the concept, this cubic function is symmetric about the origin and is an odd function: f (−x) =
− f (x).
√
4. Square Root f (x) = x Domain = {x ≥ 0} Range = {y ≥ 0}
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Chapter 1. Even and Odd Functions
This square root function is neither symmetric about the origin or the y-axis, and is neither odd nor even.
5. Absolute Value f (x) = |x| Domain = All reals
Range = {y ≥ 0}
This absolute value function is symmetric about the y-axis and is an even function: f (x) = f (−x).
6. Rational f (x) =
1
x
Domain = {x 6= 0}
Range = {y 6= 0}
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This rational value function is symmetric about the origin and is an odd function: f (−x) = − f (x).
7. Sine f (x) = sin x
Domain = All reals
Range = {−1 ≤ y ≤ 1}
This sine function is symmetric about the origin and is an odd function: f (−x) = − f (x).
8. Cosine f (x) = cos x
Domain = All reals
Range = {−1 ≤ y ≤ 1}
This sine function is symmetric about the y-axis and is an even function: f (x) = f (−x).
Practice
Is the function even, odd, or neither?
1.
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Chapter 1. Even and Odd Functions
2.
3.
4.
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Determine algebraically whether the function is even, odd or neither.
5. y = −4x
6. f (x) = 2|x|+3
7. y = (x + 3)2 − 1
8. h(x) =
x3 −2x
x4 +3x2
9. g(x) = 1 − x5 + x23
p
10. k(x) = 5x x4 − 7x2
11. y = 3 sin2 (x) − 4
12. h(x) =
sin x
x
p
|x|
p
14. f (x) = 4 |x − 1|
13. f (x) =
15. k(x) = (x + 3)(x − 4)
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Chapter 1. Even and Odd Functions
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