Chapter 3.5, Problem 23AYU

To determine

The domain of the given function is to be estimated:

  $f\left(x\right)=\sqrt{x^2-16}$

Answer to Problem 23AYU

The domain of $f\left(x\right)=\sqrt{x^2-16}$ is $\left(-\infty ,-4\right)\left(-4,4\right)\left(4,\infty \right)$ .

Explanation of Solution

Given information:

  $f\left(x\right)=\sqrt{x^2-16}$

Calculation:

The complete set of the independent variable is known as the domain of the function.

The given function is:

  $f\left(x\right)=\sqrt{x^2-16}$

The expression inside the square root cannot be negative is the domain condition.

So,

  $\begin{array}{l}{x}^2-16\ge 0\\ x^2-{\left(4\right)}^2\ge 0\\ \left(x-4\right)\left(x+4\right)\ge 0\left[\because \left(a-b\right)\left(a+b\right)=a^2-b^2\right]\end{array}$

That means there are two critical numbers as $x=-4,4$ . So, the numbers line gets divided into the three parts by the critical numbers as:

  $\left(-\infty ,-4\right)\left(-4,4\right)\left(4,\infty \right)$

So, the domain of $f\left(x\right)=\sqrt{x^2-16}$ is $\left(-\infty ,-4\right)\left(-4,4\right)\left(4,\infty \right)$

Hence, the domain of the given function is $\left(-\infty ,-4\right)\left(-4,4\right)\left(4,\infty \right)$