To find \( (f \circ g)(x) \), we need to evaluate \( f(g(x)) \).
We have the following functions:
- \( f(x) = x^2 + 2 \)
- \( g(x) = \sqrt{x + 5} \)
Now, we compute \( (f \circ g)(x) \):
\[ g(x) = \sqrt{x + 5} \]
Substituting \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(\sqrt{x + 5}) = (\sqrt{x + 5})^2 + 2 \]
Calculating this gives:
\[ (\sqrt{x + 5})^2 = x + 5 \]
Thus, we have:
\[ f(g(x)) = x + 5 + 2 = x + 7 \]
Now we need to determine the domain of \( (f \circ g)(x) \). The domain of \( g(x) \) must be considered first. The function \( g(x) = \sqrt{x + 5} \) is defined when the expression under the square root is non-negative:
\[ x + 5 \geq 0 \quad \Rightarrow \quad x \geq -5 \]
Since \( f(x) \) is a polynomial, it is defined for all real numbers.
Therefore, the domain of \( (f \circ g)(x) \) is determined by the domain of \( g(x) \), which is:
\[ [-5, \infty) \]
Now we can summarize our result:
\[ (f \circ g)(x) = x + 7; \quad \text{domain: } [-5, \infty) \]
Thus, the correct response is:
\( (f \circ g)(x) = x + 7; \quad \text{domain: } [-5, \infty) \)