To solve for [tex]\((f+g)(x)\)[/tex], we need to add the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] together.
Given:
[tex]\[ f(x) = \frac{5}{x} + 12 \][/tex]
[tex]\[ g(x) = \sqrt{x-3} + 10 \][/tex]
We then define [tex]\((f+g)(x)\)[/tex] as follows:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
Substituting the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f+g)(x) = \left( \frac{5}{x} + 12 \right) + \left( \sqrt{x-3} + 10 \right) \][/tex]
Combine the terms:
[tex]\[ (f+g)(x) = \frac{5}{x} + 12 + \sqrt{x-3} + 10 \][/tex]
[tex]\[ (f+g)(x) = \frac{5}{x} + \sqrt{x-3} + 12 + 10 \][/tex]
[tex]\[ (f+g)(x) = \frac{5}{x} + \sqrt{x-3} + 22 \][/tex]
Now, we compare this function with the given options:
A. [tex]\(\frac{5}{x} - \sqrt{x-3} + 2\)[/tex]
B. [tex]\(\frac{\sqrt{x-3} + 5}{x} + 22\)[/tex]
C. [tex]\(\frac{5}{x} + \sqrt{x} + 19\)[/tex]
D. [tex]\(\frac{5}{x} + \sqrt{x-3} + 22\)[/tex]
The correct function is:
D. [tex]\((f+g)(x) = \frac{5}{x} + \sqrt{x-3} + 22\)[/tex]
