Let's solve the problem step-by-step.
Given the function:
[tex]\[ f(x) = 3x + \frac{5}{x} \][/tex]
We need to find [tex]\( f(a+2) \)[/tex]. To do this, we substitute [tex]\( x \)[/tex] with [tex]\( a+2 \)[/tex] in the function [tex]\( f(x) \)[/tex].
So, we substitute [tex]\( a+2 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(a+2) = 3(a+2) + \frac{5}{a+2} \][/tex]
Now let's simplify this expression:
1. Distribute the 3 in [tex]\( 3(a+2) \)[/tex]:
[tex]\[ 3(a+2) = 3a + 6 \][/tex]
2. Add the second part [tex]\( \frac{5}{a+2} \)[/tex]:
[tex]\[ f(a+2) = 3a + 6 + \frac{5}{a+2} \][/tex]
So the expression for [tex]\( f(a+2) \)[/tex] simplifies to:
[tex]\[ f(a+2) = 3a + 6 + \frac{5}{a+2} \][/tex]
Let's compare this result to the given options:
A. [tex]\( 3(f(a))+\frac{5}{f(a)+2} \)[/tex]
B. [tex]\( 3(a+2)+\frac{5}{a+2} \)[/tex]
C. [tex]\( 3a+\frac{5}{a}+2 \)[/tex]
Clearly, the correct option is:
[tex]\[ \boxed{B. \, 3(a+2) + \frac{5}{a+2}} \][/tex]
Thus, [tex]\( f(a+2) = 3a + 6 + \frac{5}{a+2} \)[/tex].
The result matches option B:
[tex]\[ 3(a + 2) + \frac{5}{a + 2} \][/tex]
