Chapter 1, Problem 55CR

To determine

Tostate:State whether the given function is even, odd or neither.

Answer to Problem 55CR

The given function is an odd function.

Explanation of Solution

Given information:The given function is $f\left(x\right)=2x\sqrt{4-x^2}$

A function $y=f\left(x\right)$ is said to be an even function if the graph of the function is symmetrical about y-axis. Algebraically, $y=f\left(x\right)$ is even if $f\left(-x\right)=f\left(x\right)$ for all x in the domain.

Similarly, a function $y=f\left(x\right)$ is said to be an odd function if the graph of the function is symmetrical about origin. Algebraically, $y=f\left(x\right)$ is even if $f\left(-x\right)=-f\left(x\right)$ for all x in the domain.

Here the graph of the function is $f\left(x\right)=2x\sqrt{4-x^2}$

We have

  $\begin{array}{c}f\left(-x\right)=2\left(-x\right)\sqrt{4-{\left(-x\right)}^2}\\ =-2x\sqrt{4-x^2}\\ =-f\left(x\right)\end{array}$

Therefore, the given function is an odd function.

The graph of the function is shown below