Problem 70 - College Algebra

Question

Determine whether each function is even, odd, or neither. $$f(x)=2 x^{3}-6 x^{5}$$

Step-by-Step Solution

Answer

The function \(f(x)=2 x^{3}-6 x^{5}\) is an odd function.

Step 1: Determine if the function is even

To determine if a function is even, we substitute \(f(-x)\) for \(f(x)\) in the function equation. In this case we get \(f(-x)=2 (-x)^{3}-6 (-x)^{5} = -2x^3 + 6x^5\). This equation does not match the original function of \(f(x)=2 x^{3}-6 x^{5}\), therefore this function is not an even function.

Step 2: Determine if the function is odd

To check if a function is odd, we confirm that \(-f(x) = f(-x)\). So, \(-f(x) = - (2x^3 - 6x^5) = -2x^3 + 6x^5\). This equation matches exactly with what we obtained for \(f(-x)\) in step 1, so the function is an odd function.