Problem 69 - College Algebra

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

Step 1: Check if the function is even

Substitute \(-x\) for \(x\) in the function and simplify. If the result is the same as the original function \(f(x)=f(-x)\), then the function is even. So, we have \(f(-x)=\frac{1}{5} (-x)^{6}-3(-x)^{2}=\frac{1}{5} x^{6}-3 x^{2}=f(x)\). Therefore, the function is even.

Step 2: Check if the function is odd

Though we've already established that the function is even, we'll still check if it is odd for the sake of completion. Substitute \(-x\) for \(x\) in the function and simplify. If the result is equivalent to \(-f(x)\), then the function is odd. So, we have \(f(-x)=\frac{1}{5} (-x)^{6}-3(-x)^{2} \neq -f(x)\). Therefore, the function is not odd.