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Chapter 1: Problem 73

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

Short Answer

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

Step by step solution

01

State the function

The function given is \(f(x)=x \sqrt{1-x^{2}}\)
02

Substitute \(-x\) for \(x\)

Let's substitute \(-x\) into the function and simplify: \(f(-x)= -x \sqrt{1-(-x)^{2}} = -x \sqrt{1-x^{2}}\)
03

Compare \(f(x)\) and \(f(-x)\)

Comparing \(f(x)\) and \(f(-x)\), we have that \(f(x) = -f(-x)\). This indicates that the function is odd.
04

State the symmetry

Since the function is odd, it implies that the function is symmetric about the origin.

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