Chapter 1.1, Problem 67E

To determine

To find: Whether the given functionis even, odd, or neither.

Answer to Problem 67E

The given function is an odd function.

Explanation of Solution

Given information:

The given function is,

  $f\left(x\right)=\frac{x}{x^2+1}$

Concept used:

The function is said to be odd function if it satisfies $-f\left(x\right)=f\left(-x\right)$ for all values of $x$ and the function is said to be even function if it satisfies $f\left(x\right)=f\left(-x\right)$ for all values of $x$ .

The given function is,

  $f\left(x\right)=\frac{x}{x^2+1}$

Calculate the value of $f\left(-x\right)$ as,

  $\begin{array}{l}f\left(-x\right)=\frac{-x}{{\left(-x\right)}^2+1}\\ f\left(-x\right)=-\left[\frac{x}{x^2+1}\right]\\ f\left(-x\right)=-f\left(x\right)\end{array}$

The given function satisfies the condition of an odd function. Hence, the given function is an odd function.

Consider the graph of the given function as follows,