Determine whether the following function is even, odd, or neither.?

Question

I got an answer, but some people say that the answer is neither, and some say it is odd. So I would like someone to work it out and explain why they got neither or odd for their answer. Thank you.

The function is:

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

Please!!! Is someone going to answer????

Answer 1

The definition of even and odd function is well known.

DEFINITION. A function f is even if the graph of f is symmetric with respect to the y-axis. Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f.
A function f is odd if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if f(-x) = -f(x) for all x in the domain of f.

Let's put -x in for x in your function and see what happens.

f(-x) = -2x / (-x^3 +x)

Simplifying,

f(-x) = 2x / (x^3 - x)

which we see is identical to the original function.

The definition say that if f(-x) = f(x), the function is even, which is the case you have.

I cannot explain how "they" got odd and neither. You will have to ask them.

Answer 2

Even Odd Function Calculator

Answer 3

Even function f(-x) = f(x)
Odd function f(-x) = -f(x)
The function is even. If you replace all the x's in the equation with (-x)'s you get:
h(x) = (2(-x))/((-x)^3-(-x)) = (-2x)/(-x^3+x) = (2x)/(x^3-x), therefore f(-x) = f(x)

Example of an odd function, for comparison, would be:
h(x) = x^5 + x^3
when you replace with (-x), the becomes h(x) = -x^5 - x^3 = -(x^5 + x^3).

Answer 4

I partially agree with garypopkin. The only ambiguity resides in the value of your function h(x) for x=0.
h(x=0) = 0/0, which is indetermined. Therefore, the correct answer should be:

Even for all x except x=0; indetermined (neither even nor odd) for x=0.