Let f(x)=x^(3 )-7x. Determine if this function is even, odd or neither. Explain/Show work.

A function is even if f(-x) = f(x), while it is odd if f(-x) = -1*f(x)... These equations have to be true for all x.

If f(-x) does not equal either f(x) or -1*f(x), then it can't be called either even or odd.

If f(-x) does not equal either f(x) or -1*f(x), then it can't be called either even or odd.

So, let's evaluate f(-x):

(-x)^3 - 7*(-x) = (-1)^3*(x)^3 - (-1)*7*x

= (-1)* (x^3 - 7x)

...because (-1)*3 = -1, and that would leave a -1 that could be factored out of the subtraction...

Since f(x) = (x^3-7x), we have just shown that f(-x) = -1*f(x), and we can conclude that f(x) is odd.