Answer: [tex]-2 \le x < 5[/tex]
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Explanation:
We cannot have negative numbers under the square root, if we wanted the result to be some real number.
The stuff under the square root must be 0 or larger.
This means the x+2 must be 0 or larger.
[tex]x+2 \ge 0\\\\x \ge -2[/tex]
Similarly, the 5-x must be 0 or larger. But wait, we cannot have 0 in the denominator (or else we have a division by zero error), so 5-x must be larger than 0.
5-x > 0
5-x+x > 0+x
5+0 > 0+x
5 > x
x < 5
Combine both [tex]x \ge -2[/tex] and [tex]x < 5[/tex] to find the domain is [tex]-2 \le x < 5[/tex]
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Visual confirmation is shown below. I used Desmos which is a free graphing app.
The green curve is what we're after. It's between x = -2 and x = 5
We include -2, but exclude 5.
Take note of the closed endpoint at x = -2, and also the vertical asymptote at x = 5. The curve approaches this asymptote but never actually touches it. Think of an electric fence you can get closer to, but not actually touch.
