To solve for [tex]\( f(a+2) \)[/tex] given the function [tex]\( f(x) = \frac{3 + x}{x - 3} \)[/tex], we follow these steps:
1. Substitute [tex]\( a + 2 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(a+2) = \frac{3 + (a + 2)}{(a + 2) - 3}
\][/tex]
2. Simplify the expression inside the function:
- For the numerator: [tex]\( 3 + (a + 2) = 3 + a + 2 = a + 5 \)[/tex]
- For the denominator: [tex]\( (a + 2) - 3 = a + 2 - 3 = a - 1 \)[/tex]
3. Combine the simplified numerator and denominator:
[tex]\[
f(a+2) = \frac{a + 5}{a - 1}
\][/tex]
After substitution and simplification, we find that:
[tex]\[
f(a+2) = \frac{a + 5}{a - 1}
\][/tex]
This matches option A:
[tex]\[
\frac{5 + a}{a - 1}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{a + 5}{a - 1}}
\][/tex]
