To solve for [tex]\( f(a+2) \)[/tex] given the function [tex]\( f(x) = 3x + \frac{5}{x} \)[/tex], follow these steps:
1. Identify the Input:
We need to evaluate the function [tex]\( f(x) \)[/tex] at [tex]\( x = a + 2 \)[/tex]. This means we substitute [tex]\( a + 2 \)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex].
2. Substitute [tex]\( a+2 \)[/tex] into the Function:
[tex]\[
f(a + 2) = 3(a + 2) + \frac{5}{a + 2}
\][/tex]
3. Simplify:
First, distribute the 3 through the parentheses:
[tex]\[
f(a + 2) = 3a + 6 + \frac{5}{a + 2}
\][/tex]
4. Combine Like Terms:
Combine the terms to get the final expression for [tex]\( f(a+2) \)[/tex]:
[tex]\[
f(a + 2) = 3a + 6 + \frac{5}{a + 2}
\][/tex]
5. Evaluate for a Specific [tex]\( a \)[/tex]:
If necessary, you can evaluate this expression for specific values of [tex]\( a \)[/tex]. However, based on our earlier substitution and simplification, the result when [tex]\( a = 2 \)[/tex] is:
[tex]\[
f(2+2) = 13.25
\][/tex]
In summary, the function value [tex]\( f(a+2) \)[/tex] given [tex]\( f(x) = 3x + \frac{5}{x} \)[/tex] indeed simplifies to [tex]\( 3a + 6 + \frac{5}{a + 2} \)[/tex]. When evaluated at [tex]\( a = 2 \)[/tex], it results in [tex]\( 13.25 \)[/tex].
