Problem 28 - 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 provided function \(f(x) = x \sqrt{1-x^{2}}\) is an odd function.

Step 1: Evaluate the function at -x

The first step in solving this problem is to replace x in the function with -x. Therefore, substitute -x into the function \(f(x)=x\sqrt{1-x^{2}}\) to get \(f(-x) = -x\sqrt{1-(-x)^{2}}\).

Step 2: Simplify the function at -x

By simplifying the function at -x, we get \(f(-x) = -x\sqrt{1-x^{2}}\). The squared part of the function takes away the negative sign and leaves the function unchanged.

Step 3: Evaluate if the function is even, odd, or neither

We can see that \(f(-x) = -f(x)\), indicating that for every x in the domain of f, f(-x) is equal to -f(x). Hence the function f(x) is an odd function.