Problem 71 - College Algebra

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: Write down the given function

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

Step 2: Substitute \(-x\) for \(x\) in the given function

We substitute \(-x\) for \(x\) in the given function: \(f(-x) = -x \sqrt{1-(-x)^{2}} = -x \sqrt{1 - x^{2}}\)

Step 3: Compare the obtained function with the original function

Comparing \(f(-x) = -x \sqrt{1 - x^{2}}\) with \(f(x) = x \sqrt{1 - x^{2}}\) we notice that \(f(-x)\) is not equal to \(f(x)\), so the function is not even. However, \(f(-x)\) is equal to \(-f(x)\), so the function is odd.