Even and Odd Functions
Table of Contents
Definitions of Even & Odd Functions 2
Algebraic Definition 2
Graphic Definition 4
Combining Even & Odd Functions 6
Multiplication 6
Addition 7
Integrals of Even & Odd Functions 7
Fourier Series: Even & Odd Functions 9
Arbitrary Period (2L) 9
Case of Period 2π 10
References 14
Algebraic Definitions
1) Even Function:
2) Odd Function:
Algebraically
You may be asked to "determine algebraically" whether a function is even or odd. To do this, you take the function and plug –x in for x, and then simplify. If you end up with the exact same function that you started with (that is, if f(–x) = f(x), so all of the signs are the same), then the function is even. If you end up with the exact opposite of what you started with (that is, if f(–x) = –f(x), so all of the "plus" signs become "minus" signs, and vice versa), then the function is odd. In all other cases, the function is "neither even nor odd".
Example 1:
Determine algebraically whether f(x) = –3x2 + 4 is even, odd, or neither.
So I 'll plug –x in for x, and simplify: f(–x) = –3(–x)2 + 4 = –3(x2) + 4 = –3x2 + 4 = f(x)
My final expression is the same thing I 'd started with, which means that f(x) is even.
Example 2:
Example 3:
Determine algebraically whether f(x) = 2x3 – 4x is even, odd, or neither.
I 'll plug –x in for x, and simplify: f(–x) = 2(–x)3 – 4(–x) = 2(–x3) + 4x = –2x3 + 4x
My final expression is the exact opposite of what I started with, by which I mean that the sign on each term has been changed to its opposite, just as if I 'd multiplied through by –1:
–f(x) = –1[f(x)] = –[2x3 – 4x] = –2x3 + 4x
This means that f(x) is odd.
Example 4:
Determine algebraically whether f(x) = 2x3 – 3x2 – 4x + 4 is even, odd, or neither.
I"ll plug –x in for x, and simplify:
f(–x) = 2(–x)3 – 3(–x)2 – 4(–x) + 4 = 2(–x3) – 3(x2) + 4x + 4 = –2x3 –3x2 +
Definitions of Even & Odd Functions 2
Algebraic Definition 2
Graphic Definition 4
Combining Even & Odd Functions 6
Multiplication 6
Addition 7
Integrals of Even & Odd Functions 7
Fourier Series: Even & Odd Functions 9
Arbitrary Period (2L) 9
Case of Period 2π 10
References 14
Algebraic Definitions
1) Even Function:
2) Odd Function:
Algebraically
You may be asked to "determine algebraically" whether a function is even or odd. To do this, you take the function and plug –x in for x, and then simplify. If you end up with the exact same function that you started with (that is, if f(–x) = f(x), so all of the signs are the same), then the function is even. If you end up with the exact opposite of what you started with (that is, if f(–x) = –f(x), so all of the "plus" signs become "minus" signs, and vice versa), then the function is odd. In all other cases, the function is "neither even nor odd".
Example 1:
Determine algebraically whether f(x) = –3x2 + 4 is even, odd, or neither.
So I 'll plug –x in for x, and simplify: f(–x) = –3(–x)2 + 4 = –3(x2) + 4 = –3x2 + 4 = f(x)
My final expression is the same thing I 'd started with, which means that f(x) is even.
Example 2:
Example 3:
Determine algebraically whether f(x) = 2x3 – 4x is even, odd, or neither.
I 'll plug –x in for x, and simplify: f(–x) = 2(–x)3 – 4(–x) = 2(–x3) + 4x = –2x3 + 4x
My final expression is the exact opposite of what I started with, by which I mean that the sign on each term has been changed to its opposite, just as if I 'd multiplied through by –1:
–f(x) = –1[f(x)] = –[2x3 – 4x] = –2x3 + 4x
This means that f(x) is odd.
Example 4:
Determine algebraically whether f(x) = 2x3 – 3x2 – 4x + 4 is even, odd, or neither.
I"ll plug –x in for x, and simplify:
f(–x) = 2(–x)3 – 3(–x)2 – 4(–x) + 4 = 2(–x3) – 3(x2) + 4x + 4 = –2x3 –3x2 +
References: 1) www2.kau.se/yourshes/AB2_12.pdf 2) http://www.intmath.com/fourier-series/3-fourier-even-odd-functions.php. 3) www.sakshieducation.com/.../MathMethods-Fourier_Series.pdf 4) Advanced Engineering Mathematics (by Erwin Kreyszig 9th Edition) 5) www.sunlightd.com/Fourier/FourierExpansions.aspx 6) archives.math.utk.edu/visual.calculus/0/functions.14/index.html 7) www.purplemath.com/modules/fcnnot3.htm ---------------------------------------