To find the function [tex]\((f+g)(x)\)[/tex], we need to combine the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] by adding them together. Let's add the expressions given for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
Given:
[tex]\[ f(x) = \frac{5}{x} + 12 \][/tex]
[tex]\[ g(x) = \sqrt{x-3} + 10 \][/tex]
To find [tex]\( (f+g)(x) \)[/tex], we add the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
Substituting the given functions into this expression:
[tex]\[ (f+g)(x) = \left(\frac{5}{x} + 12\right) + \left(\sqrt{x-3} + 10\right) \][/tex]
Now, combine the terms:
[tex]\[ (f+g)(x) = \frac{5}{x} + 12 + \sqrt{x-3} + 10 \][/tex]
Combine the constants [tex]\(12\)[/tex] and [tex]\(10\)[/tex]:
[tex]\[ (f+g)(x) = \frac{5}{x} + \sqrt{x-3} + 22 \][/tex]
Thus, the correct function is:
[tex]\[ (f+g)(x) = \frac{5}{x} + \sqrt{x-3} + 22 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{C. \ (f+g)(x) = \frac{5}{x} + \sqrt{x-3} + 22} \][/tex]
