Even and Odd Functions
Look at the graphs of the two functions f(x) = x^{2}  18 and g(x) = x^{3}  3x. The function f(x) = x^{2}  18 is symmetric with respect to the yaxis and is thus an even function. The function g(x) = x^{3}  3x is symmetric about the origin and is thus an odd function.
Stated another way, functions are even if changing x to x does not change The value of the function.
EVEN FUNCTION:
f(x) = x^{2}  18f(x) = (x)^{2}  18 = x^{2}  18
Since f(x) = f(x) the function is even.
Functions are odd if changing x to x negates the value of the function.
ODD FUNCTION:
f(x) = x^{3}  3xf(x) = (x)^{3}  3(x) = x^{3} + 3x = (x^{3}  3x)
Since f(x) = f(x) the function is odd.
A function can be even, odd or neither even nor odd. To determine if a function has even or odd symmetry use the following guidelines.
GUIDELINES TO DETERMINE EVEN OR ODD SYMMETRY
2. Compare the results of step 1 to f(x) and f(x).
3. Determine if the function is even, odd or neither
b. If f(x) results in the same value as f(x), the function is odd.
c. If f(x) did not result in step 4 or 5, the function is neither even nor odd.
Let's look at a couple examples.
Step 1: Replace f(x) with f(x) and simplify the function. 
Original function: f(x) = x^{6} + 4x^{2}  1 Substitute x with x: f(x) = (x)^{6} + 4(x)^{2}  1 Simplify: f(x) = x^{6} + 4x^{2}  1 
Step 2: Compare f(x) to f(x) and f(x). 
f(x) = x^{6} + 4x^{2}  1 f(x) = x^{6} + 4x^{2}  1 Compare f(x) to f(x): ${x}^{6}+4{x}^{2}1{=}{x}^{6}+4{x}^{2}1$ Since f(x) to f(x) there is no need to compare f(x) to f(x) because the function cannot be both. 
Step 3: Determine if the function is even, odd, or neither. 
Since f(x) = f(x) the function is even and has symmetry about the yaxis. 
Step 4: Graph the function

Step 1: Replace f(x) with f(x) and simplify the function. 
Original function: f(x) = x^{3} + 2x^{2}  x Substitute x with x: f(x) = (x)^{3} + 2(x)^{2}  (x) Simplify: f(x) = x^{3} + 2x^{2} + x 
Step 2: Compare f(x) to f(x) and f(x). 
f(x) = (x^{3}  2x^{2}  x) f(x) = x^{3} + 2x^{2}  x f(x) = (x^{3} + 2x^{2}  x) Compare f(x) to f(x): $({x}^{3}2{x}^{2}x){\ne}{x}^{3}+2{x}^{2}x$ Compare f(x) to f(x): $({x}^{3}2{x}^{2}x){\ne}\left({x}^{3}+2{x}^{2}x\right)$ 
Step 3: Determine if the function is even, odd, or neither. 
Since f(x) does not equal f(x) or f(x) the function is neither even nor odd. 
Step 4: Graph the function

Related Links: Math algebra Function Transformations Rolle's Theorem Calculus Topics 
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