Chapter 1, Problem 86RE

To determine

To state : that the function is even, odd or neither.

Answer to Problem 86RE

The function is odd.

Explanation of Solution

Given information :

The function is $f\left(x\right)=x\sqrt{x^2-1}$.

Formula used :

The function is even if $f\left(x\right)=f\left(-x\right)$.

The function is odd if $f\left(x\right)=-f\left(-x\right)$

Calculation : to state that the function is even, odd or neither substitute $x=-x$ in the function,

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

Thus, the function is odd.

By using the graphing utility,

The graph can be observed as:

  

From the above graph it can be observed that the functionis odd.