Select the correct answer.

If [tex]$f(x) = \frac{2x + 1}{x - 4}$[/tex], what is the value of [tex]$f^{-1}(3)$[/tex]?

A. [tex][tex]$\frac{22}{7}$[/tex][/tex]

B. [tex]$\frac{19}{11}$[/tex]

C. [tex]$\frac{8}{3}$[/tex]

D. 13

E. 11



Answer :

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]