To find the value of [tex]\( f^{-1}(3) \)[/tex] for the function [tex]\( f(x) = \frac{2x+1}{x-4} \)[/tex], we need to find the value of [tex]\( x \)[/tex] such that [tex]\( f(x) = 3 \)[/tex].
1. Start with the equation [tex]\( f(x) = 3 \)[/tex]:
[tex]\[
\frac{2x+1}{x-4} = 3
\][/tex]
2. To eliminate the fraction, multiply both sides by [tex]\( x - 4 \)[/tex]:
[tex]\[
2x + 1 = 3(x - 4)
\][/tex]
3. Distribute the 3 on the right side of the equation:
[tex]\[
2x + 1 = 3x - 12
\][/tex]
4. Subtract [tex]\( 2x \)[/tex] from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
1 = x - 12
\][/tex]
5. Add 12 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 13
\][/tex]
So, the value of [tex]\( f^{-1}(3) \)[/tex] is [tex]\( 13 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{13} \][/tex]
