## College Algebra (10th Edition)

$\color{blue}{[-1, 2) \cup (2, +\infty)}$
Factor the denominator of the function to obtain: $f(x) = \dfrac{\sqrt{x+1}}{(x+2)(x-2)}$ The value of $x$ can be any real number except the ones that will make the function either undefined or have an imaginary number value. The given function is undefined when the value of the denominator is zero. This is because division of zero leads to an undefined expression. Find the values of $x$ that will make the denominator zero by equating each factor of the denominator to zero and then solving each equation. $\begin{array}{ccc} &x+2=0 &\text{ or } &x-2=0 \\&x=-2 &\text{ or } &x=2 \end{array}$ The function will have an imaginary number value when the radicand (expression inside the radical sign) is negative. This means that the radicand must be greater than or equal to zero. $x+1 \ge 0 \\x\ge 0-1 \\x \ge -1$ Thus, the domain of the given function are the numbers that satisfy the following conditions: $x \ge -1$ $x \ne -2$ $x \ne 2$. Therefore, the domain of the given function in interval notation is: $\color{blue}{[-1, 2) \cup (2, +\infty)}$