Problem 27 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The function \(f(x) = x\sqrt{1-x^2}\) is odd.

Step 1: Substitute \(x\) for \(-x\)

Replace all occurences of \(x\) in the original function with \(-x\). This gives: \(f(-x) = -x\sqrt{1-(-x)^2} = -x\sqrt{1-x^2}\)

Step 2: Compare \(f(-x)\) with \(f(x)\) and \(-f(x)\)

We now have to analyse our function \(f(-x) = -x\sqrt{1-x^2}\). Clearly, this is not equal to \(f(x) = x\sqrt{1-x^2}\) so the function is not even. However, multiply \(f(x)\) by -1 to get \(-f(x) = -x\sqrt{1-x^2}\), which is now identical to \(f(-x)\) so the function is odd.