Answer
$[-1, 2) \cup (2, +\infty)$
Work Step by Step
We will factor the denominator of the function to obtain:
$f(x) = \dfrac{\sqrt{x+1}}{(x+2)(x-2)}$
We observe that the function is undefined when the denominator becomes $0$ because division by zero makes the function undefined.
In order to compute the domain of the given function, we will first exclude those values of $x$ that will make the function undefined. The square root function is undefined for negative numbers because the square root of a negative number is an imaginary number.
We will equate each factor of the denominator to zero and then simplifying each equation as follows:
$x+2 =0 \implies x=-2$
and
$x-2 =0 \implies x=2$
Next, the radicand must be greater than or equal to zero.
$x+1 \ge 0 \implies x \ge -1$
Thus, the following conditions satisfy the function:
$x \ge -1 \\ x \ne -2 \\ x \ne 2$.
So, we write the domain of the given function as:
$[-1, 2) \cup (2, +\infty)$
