Chapter 1.2, Problem 14E

Solution

To determine: the domain of the function algebraically.

The domain of the function is $\left[-2,2\right]$ .

Given Information:

Consider the function $\frac{\sqrt{4-x^2}}{x-3}$ .

Concept used:

Set the denominator of function equals to zero to find the domain of the function.

Calculation:

For the domain of the function, the numerator has to be non-negative and denominator to be non-zero since the denominator can’t be zero.

Here the square root cannot be negative.

Simplify the numerator of the function $\frac{\sqrt{4-x^2}}{x-3}$ as follows.

  $\begin{array}{c}4-x^2\ge 0\\ -x^2\ge -4\\ x^2\le -4\\ -2<x<2\end{array}$

Set the denominator equal to zero and simplify as follows.

  $\begin{array}{c}x-3=0\\ x=3\end{array}$

Therefore, the domain of the function is $\left[-2,2\right]$ .