Question

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Answer 1

The first thing you should do in this case is to compose both functions:
 f (x) = x + 7
 g (x) = 1 / (x-13)
 Making the composition
 f (g (x)) = (1 / (x-13)) + 7
 f (g (x)) = (7 (x-13) +1) / (x-13)
 Answer:
 The domain of the function is:
 x other than 13
 (option 4)

Answer 2

The domain of a composite function $f(g(x))$ is, technically speaking, the set of those inputs $x$ in the domain of $g$ for which $g(x)$ is in the domain of $f$. What this is really saying, though, is that the domain is for those $x$ that overlap in both functions, or that are defined for both $f(x)$ and $g(x)$. You can write {$x \ | \ D_f$ ∩ $D_g$}.

For this specific example, the domain of $f(x)$ is {all real numbers $x$}. The domain of $g(x)$ is {$x \ | \ x \neq 13$}. These overlap for all $x$ such that $x \neq 13$, so the domain of $f(g(x))$ is {$x \ | \ x \neq 13$}.