Problem 25 - College Algebra Essentials

Question

Determine whether each function is even, odd, or neither. $$f(x)=\frac{1}{5} x^{6}-3 x^{2}$$

Step-by-Step Solution

Answer

The given function \(f(x)=\frac{1}{5} x^{6}-3 x^{2}\) is even and it is not odd.

Step 1: Determine if the function is even

Replace \(x\) with \(-x\) in the function to see if the original function is obtained. So, \(f(-x)=\frac{1}{5} (-x)^{6}-3 (-x)^{2}\). This simplifies to \(f(-x)=\frac{1}{5} x^{6}-3 x^{2}\), which is the same as the original function. Therefore, the function is even.

Step 2: Determine if the function is odd

Replace \(x\) with \(-x\) in the function and negate it to see if the original function is obtained. So, \(-f(-x)= -\left[\frac{1}{5} (-x)^{6}-3(-x)^{2}\right]\). This simplifies to \(-f(-x)= -\frac{1}{5} x^{6} +3 x^{2}\), which is not the same as the original function. Therefore, the function is not odd.