Problem 47 - Precalculus Essentials

Question

Determine whether each function is even, odd, or neither. Then determine whether the function's graph is symmetric with respect to the \(y\) -axis, the origin, 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, and its graph is symmetric about the origin.

Step 1: Identify the function

Given the function \(f(x) = x \sqrt{1 - x^{2}}\).

Step 2: Test for even function

An even function is such that \(f(x) = f(-x)\). Let’s replace \(x\) with \(-x\) in our function and see if the original function is obtained: \(f(-x) = -x \sqrt{1 - (-x)^{2}} = -x \sqrt{1 - x^{2}}\). This is not equal to the original function \(f(x)\). Thus, the function is not even.

Step 3: Test for odd function

An odd function is such that \(f(x) = -f(-x)\). We essentially performed this test in step 2, as we found that \(f(-x)\) equals \(-f(x)\). Thus, the function is odd.

Step 4: Test for symmetry

Since the function is odd, its graph is symmetric about the origin.