Determine algebraically whether each function is even, odd or neither.
(1) f(x) = 2x^4 - x^2
(2) y = x/(x^2 - 1)
Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all x in the domain of f:
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Again, let f(x) be a real-valued function of a real variable. Then f is odd if the following equation holds for all x in the domain of f:
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(1) $\displaystyle f(x)=2x^4-x^2$
$\displaystyle f(x)=2x^4-x^2=f(-x)=2(-x)^4-(-x)^2=2x^4-x^2$
$\displaystyle f(x)=f(-x)$. Therefore, this function is even.
(2)$\displaystyle y=\frac{x}{x^2-1}$
The quotient of two even functions is an even function.
The quotient of two odd functions is an even function.
The quotient of an even function and an odd function is an odd function.
See here: Function Notation: Even and Odd Functions
for a pretty good explanation. The first paragraph pretty much lays it out.
".....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)), 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)), then the function is odd. In all other cases, the function is neither even nor odd."
See example 3.
For example, $\displaystyle f(x)= x^2+ x$ is neither even nor odd. f(1)= 1+1= 2 while f(-1)= 1- 1= 0. Neither f(-x)= f(x) nor f(-x)= -f(x) is true.
It's pretty easy to show that all polynomials with only even powers are even functions, all polynomials with only odd powers are odd functions and that polynomials with both even and odd powers are neither. The rule masters gave is makes it easy to see when rational functions are even or odd but "even" and "odd" applies to other functions as well. sin(x) is an odd function and cos(x) is an even function.