I'm interpretting your function to be:
[tex]f(x) = \dfrac{\sqrt{2+x}}{3-x}[/tex]
Going with that function, there are two rules to keep in mind when working with domains of functions:
Looking at Rule #1: Do not divide by zero...
You need to ask yourself, "What value of x would make that denominator equal 0? What does [tex]3-x =0[/tex]?"
The answer is 3. [tex]x=3[/tex] makes the denominator 0, so 3 CANNOT be in the domain. Every other real number is fine for the denominator.
Looking at Rule #2: Avoid square roots of negative numbers...
You want to be sure what you take the square root of is NOT negative, so you want to be sure it IS zero or positive.
You need to solve the inequality [tex]2+x \ge 0[/tex].
That will tell you want numbers DO work for the square root.
[tex]\begin{aligned} 2+x &\ge 0 \\[0.5em]2 -2+x &\ge 0-2\\[0.5em]x &\ge -2\end{aligned}[/tex]
So, as long as your x-values are -2 or greater, you're good in terms of the square root not having any issue.
Putting the two rules together...
You need to include all the numbers from Rule #2, except for those that were a problem from Rule #1.
What do you get when you put those together?
