To determine [tex]\( f(a+2) \)[/tex] for the function [tex]\( f(x) = 3x + \frac{5}{x} \)[/tex], we need to follow these steps:
1. Identify the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 3x + \frac{5}{x}
\][/tex]
2. Substitute [tex]\( a+2 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(a+2) = 3(a+2) + \frac{5}{a+2}
\][/tex]
3. Simplify the expression:
- Distribute 3 within the parentheses:
[tex]\[
3(a+2) = 3a + 6
\][/tex]
- Combine this result with the fraction:
[tex]\[
f(a+2) = 3a + 6 + \frac{5}{a+2}
\][/tex]
Thus, after substituting [tex]\( a+2 \)[/tex] into the function and simplifying, we arrive at:
[tex]\[
f(a+2) = 3a + 6 + \frac{5}{a+2}
\][/tex]
Comparing this result with the given options:
A. [tex]\( 3(a+2) + \frac{5}{a+2} \)[/tex]
B. [tex]\( 3(f(a)) + \frac{5}{f(a) + 2} \)[/tex]
C. [tex]\( 3a + \frac{5}{6} + 2 \)[/tex]
Option A matches our result after substitution:
[tex]\[
3(a+2) + \frac{5}{a+2} = 3a + 6 + \frac{5}{a+2}
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
