Answer: choice A
[tex]D: x > -3; \ (f \circ g)(x) = \frac{-7}{\sqrt{x+3}}[/tex]
==================================
Explanation:
The idea is to start with f(x), then replace every copy of x with g(x) so that you end up with the composition function as shown below.
[tex]f(x) = \frac{-7}{x}\\\\f(g(x)) = \frac{-7}{g(x)}\\\\f(g(x)) = \frac{-7}{\sqrt{x+3}}\\\\(f \circ g)(x) = \frac{-7}{\sqrt{x+3}}\\\\[/tex]
The notation [tex]f(g(x))[/tex] is the same as [tex](f \circ g)(x)[/tex]
I prefer [tex]f(g(x))[/tex] as it is more descriptive.
--------
Now onto computing the domain. We want to avoid dividing by zero, and we want to avoid taking the square root of a negative number.
This means the expression x+3 must be larger than 0. That way [tex]\sqrt{x+3}[/tex] will also be larger than 0.
If x+3 > 0, then x > -3 after subtracting 3 from both sides.
As long as x is larger than -3, then it is a valid input to the composite function [tex](f \circ g)(x)[/tex]
