Even, odd, or neither

Question

How do you determine?

Determine algebraically whether the given function is even, odd, or neither.

F(x)= -x^3/ 3x^2-9

Answer 1

As the numerator is odd (only odd powers of x), and the denominator is even (only even), the fraction is odd. This is easy to prove.

Consider h(x) = f(x)/g(x)

f(-x) = - f(x) (an odd function)

g(-x) = g(x)

Then, h(-x) = f(-x)/g(-x) = -f(x)/g(x) = -(f(x)/g(x)) = - h(x)

Thus, h(-x) = - h(x)

h is odd.