To solve for [tex]\( f(a+2) \)[/tex] given the function [tex]\( f(x) = 3x + \frac{5}{x} \)[/tex], we need to substitute [tex]\( x \)[/tex] with [tex]\( a+2 \)[/tex] in the function.
1. Start with the given function:
[tex]\[
f(x) = 3x + \frac{5}{x}
\][/tex]
2. Substitute [tex]\( x \)[/tex] with [tex]\( a+2 \)[/tex]:
[tex]\[
f(a+2) = 3(a+2) + \frac{5}{a+2}
\][/tex]
3. Simplify the expression:
[tex]\[
f(a+2) = 3(a+2) + \frac{5}{a+2}
\][/tex]
Thus, the expression for [tex]\( f(a+2) \)[/tex] is:
[tex]\[
f(a+2) = 3(a+2) + \frac{5}{a+2}
\][/tex]
Now let's check the given options:
- Option A: [tex]\( 3(a+2) + \frac{5}{a+2} \)[/tex]
This matches our derived expression for [tex]\( f(a+2) \)[/tex], so Option A is correct.
- Option B: [tex]\( 3(f(a)) + \frac{5}{f(a) + 2} \)[/tex]
To check this, let's first find [tex]\( f(a) \)[/tex]:
[tex]\[
f(a) = 3a + \frac{5}{a}
\][/tex]
Now, substituting into Option B:
[tex]\[
3(f(a)) + \frac{5}{f(a) + 2} = 3 \left( 3a + \frac{5}{a} \right) + \frac{5}{3a + \frac{5}{a} + 2}
\][/tex]
This is a completely different expression and is not equivalent to [tex]\( f(a+2) \)[/tex].
- Option C: [tex]\( 3a + \frac{5}{a} + 2 \)[/tex]
This suggests that [tex]\( f(a+2) = f(a) + 2 \)[/tex], which is also not correct based on the function definition.
Therefore, the correct answer is:
[tex]\[
\boxed{\text{A. } 3(a+2) + \frac{5}{a+2}}
\][/tex]
