Chapter 1.5, Problem 73E

To determine

To calculate; function is even, odd or neither and then describe the symmetry.

Answer to Problem 73E

Solution: Odd function, symmetric about origin.

Explanation of Solution

  $f(x)=x\sqrt{1-x^2}$

Let

  $y=x\sqrt{1-x^2}$

Compute $f(-x)$ , because if $f(x)=-f(x)$ , then the function is odd.

  $\begin{array}{l}f(-x)=-x\sqrt{1-{(-x)}^2}\\ =-x\sqrt{1-x^2}\\ =-f(x)\end{array}$

Hence, the function is odd.

The symmetry about the origin is checked by replacing $x$ by $-x.$ and $y$ by $-y.$

  $\begin{array}{l}-y=-x\sqrt{1-{(-x)}^2}\\ y=x\sqrt{1-x^2}\end{array}$

The result is equivalent equation, hence symmetry about origin.