# Even and Odd Functions

Table of Contents Definitions of Even & Odd Functions Algebraic Definition 2 Graphic Definition 4 2 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 21110 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 igns are the same), then the function is even. If you end up with the exact opposite of what you started with (that is, iff(-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) = -3×2 + 4 is even, odd, or neither.

So I’ll plug -x in for x, and simplify: ??” ??”3(x2) + 4 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) = 2×3 – 4x is even, odd, or neither. I’ll plug -x in for x, and simplify: = 2(-X3) + 4X = -213 + 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) – -1213 – 4X] = ??”2×3 + 4x This means that f(x) is odd.Example 4: Determine algebraically whether f(x) plug -x in for x, and simplify: = ??” 3(X2) + 4X + 4 = ??”213 ??”312 + 4x + 4 = 213 – 3×2 -4x+ 4 is even, odd, or neither. ‘”II This is neither the same thing I started with (namely, 2×3 – 3×2 – 4x + 4) nor the exact pposite of what I started with (namely, -2×3 + 3×2 + 4x – 4). This means that f(x) is neither even nor odd. Graphic Definitions These ideas are best illustrated using some basic functions.Let’s see a few examples for more clarification: Example 1: The diagram shows the graph of f(X) = loeu Combining even and odd functions 1) Multiplication Example 1: 2) Addition Integrals: Even and Odd Functions Example 1: (Even Function) 2{1-0} {16-16} Graph of y = 5×3 (Odd Function) Fourier Series: Even & Odd Functions 1) Even Functions: 2) Odd Functions For a Period of 211 Solution:

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