To determine [tex]\( f(a+2) \)[/tex] for the function [tex]\( f(x) = 3x + \frac{5}{x} \)[/tex], let's proceed step-by-step.
1. Substitute [tex]\(a+2\)[/tex] for [tex]\(x\)[/tex] in the function [tex]\( f(x) \)[/tex]:
Given:
[tex]\[
f(x) = 3x + \frac{5}{x}
\][/tex]
We need to find [tex]\( f(a+2) \)[/tex]. So substitute [tex]\(a+2\)[/tex] for [tex]\(x\)[/tex] in [tex]\( f(x) \)[/tex]:
[tex]\[
f(a+2) = 3(a+2) + \frac{5}{a+2}
\][/tex]
2. Simplify the expression:
- First, distribute the 3 inside the parentheses:
[tex]\[
3(a+2) = 3a + 6
\][/tex]
- The term [tex]\(\frac{5}{a+2}\)[/tex] remains as is.
3. Combine the terms:
So the entire expression for [tex]\( f(a+2) \)[/tex] becomes:
[tex]\[
f(a+2) = 3a + 6 + \frac{5}{a+2}
\][/tex]
This is our final, simplified expression.
Given the choices:
- A: [tex]\( 3(a+2) + \frac{5}{a+2} \)[/tex]
- B: [tex]\( 3a + \frac{5}{a} + 2 \)[/tex]
- C: [tex]\( 3(f(a)) + \frac{5}{f(a) + 2} \)[/tex]
The correct answer is:
A. [tex]\( 3(a+2) + \frac{5}{a+2} \)[/tex]
However, simplifying this result further as we did above should match the expression:
[tex]\[
3a + 6 + \frac{5}{a+2}
\][/tex]
This confirms our answer.
