Chapter 11, Problem 7CR

To determine

To find: Whether the function $g\left(x\right)=\frac{2x^3}{x^4+1}$ is even or odd.

Answer to Problem 7CR

The function is odd.

Explanation of Solution

Given information:

The given equation is $g\left(x\right)=\frac{2x^3}{x^4+1}$

Calculation:

Calculate the function,

  $\begin{array}{c}g\left(-x\right)=\frac{2{\left(-x\right)}^3}{{\left(-x\right)}^4+1}\\ =-\frac{2x^3}{x^4+1}\\ =-g\left(x\right)\end{array}$

Since, $g\left(-x\right)=-g\left(x\right)$ , the given function is odd.

A function will be symmetric with respect to the y-axis only when the function is even. When the function is odd, it will be symmetric with respect to the origin.

Therefore, the given function is odd, it is symmetric with respect to the origin.