Problem 72 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Substitute x with -x in the function

First, substitute \(x\) with \(-x\) in the function \(f(x) = x^{2}\sqrt{1 - x^{2}}\), we get \(f(-x) = (-x)^{2}\sqrt{1 - (-x)^{2}}\). By simplifying, we get \(f(-x) = x^{2}\sqrt{1 - x^{2}}\).

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

Now, it's time to compare \(f(-x) = x^{2}\sqrt{1 - x^{2}}\) and \(f(x) = x^{2}\sqrt{1 - x^{2}}\). As they are both identical, we can conclude that f(x) is an even function.