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Chapter 1: Problem 47

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. $$f(x)=x \sqrt{1-x^{2}}$$

Short Answer

Expert verified
The function \(f(x) = x \sqrt{1 - x^{2}}\) is odd, and its graph is symmetric about the origin.

Step by step solution

01

Identify the function

Given the function \(f(x) = x \sqrt{1 - x^{2}}\).
02

Test for even function

An even function is such that \(f(x) = f(-x)\). Let鈥檚 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.
03

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.
04

Test for symmetry

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

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