How do you determine if h(x)= (2x)/(x^3 - x) is an even or odd function?

1 Answer
Apr 8, 2016

Simplify and analyse h(x) to find that it is an even function.

Explanation:

h(x) = (2x)/(x^3-x) = 2/(x^2-1)

with exclusion x != 0

Since (-x)^2 = x^2, we find h(-x) = h(x) for all x in the domain of h(x)

So h(x) is an even function.

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Another quick method of finding that this is an even function is to look at the numerator and denominator polynomials.

They both consist solely of terms with odd degree. So h(x) is a quotient of two odd functions, so is an even function.

To see that the quotient of any two odd functions is an even function, suppose that f(x) and g(x) are both odd functions.

By definition f(-x) = -f(x) and g(-x) = -g(-x) for all x.

So we find:

f(-x)/g(-x) = (-f(x))/(-g(x)) = f(x)/g(x) for all x