Chapter 1, Problem 26RE

To determine

To check: Whether the function $y=\sqrt{x^4-1}$ is even, odd or neither.

Answer to Problem 26RE

The function $y=\sqrt{x^4-1}$ is an even function.

Explanation of Solution

Given information: The given function is $y=\sqrt{x^4-1}$ .

Calculation:

Even function: A function $f\left(x\right)$ is said to be an even function if it satisfies the condition $f\left(-x\right)=f\left(x\right)$ .

Odd function: A function $f\left(x\right)$ is said to be an odd function if it satisfies the condition $f\left(-x\right)=-f\left(x\right)$ .

Substitute $-x$ for $x$ in the given function $y=f\left(x\right)=\sqrt{x^4-1}$ .

  $\begin{array}{c}f\left(-x\right)=\sqrt{{\left(-x\right)}^4-1}\\ =\sqrt{x^4-1}\\ =f\left(x\right)\end{array}$

Therefore, the function $y=\sqrt{x^4-1}$ is an even function.