Problem 18 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: Testing for an Even Function

Replace \(x\) in the function \(f(x)=x^{3}-x\) with \(-x\) to see if we get \(f(x)\). The new function becomes \(f(-x)=(-x)^{3}-(-x)= -x^{3} + x\). This is not equal to the original function, \(f(x)\), so the function is not even.

Step 2: Testing for an Odd Function

Now, to check if the function is odd, we need to see if \(f(-x)\) is equal to \(-f(x)\). Taking the negative of the original function, we get \(-f(x)=-x^{3}+x\). Comparing this to \(f(-x)\) which is \(-x^{3}+x\), they are the same, implying that the function is odd.