To determine which function defines [tex]\((f+g)(x)\)[/tex], we start by understanding the operations of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] individually.
Given:
[tex]\[ f(x) = \frac{5}{x} + 12 \][/tex]
[tex]\[ g(x) = \sqrt{x-3} + 10 \][/tex]
The composition [tex]\((f+g)(x)\)[/tex] is the sum of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f+g)(x) = f(x) + g(x)
\][/tex]
Step-by-step:
1. Add [tex]\(f(x)\)[/tex] to [tex]\(g(x)\)[/tex]:
[tex]\[ f(x) = \frac{5}{x} + 12 \][/tex]
[tex]\[ g(x) = \sqrt{x-3} + 10 \][/tex]
So,
[tex]\[
(f+g)(x) = \left( \frac{5}{x} + 12 \right) + \left( \sqrt{x-3} + 10 \right)
\][/tex]
2. Combine like terms:
[tex]\[
(f+g)(x) = \frac{5}{x} + 12 + \sqrt{x-3} + 10
\][/tex]
Simplify this further:
[tex]\[
(f+g)(x) = \frac{5}{x} + \sqrt{x-3} + 22
\][/tex]
Thus, the correct function that defines [tex]\((f+g)(x)\)[/tex] is:
[tex]\[
D. \left( f+g \right)(x) = \frac{5}{x} + \sqrt{x-3} + 22
\][/tex]
