If [tex]f(x)[/tex] and [tex]f^{-1}(x)[/tex] are inverse functions of each other and [tex]f(x)=2x+5[/tex], what is [tex]f^{-1}(8)[/tex]?

A. [tex]-1[/tex]
B. [tex]\frac{3}{2}[/tex]
C. [tex]\frac{41}{8}[/tex]
D. [tex]23[/tex]



Answer :

To find the inverse function [tex]\( f^{-1}(x) \)[/tex] for the given function [tex]\( f(x) = 2x + 5 \)[/tex], follow these steps:

1. Start with the function [tex]\( f(x) = 2x + 5 \)[/tex]. Replace [tex]\( f(x) \)[/tex] by [tex]\( y \)[/tex]
[tex]\[ y = 2x + 5 \][/tex]

2. Swap the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
[tex]\[ x = 2y + 5 \][/tex]

3. Solve for [tex]\( y \)[/tex]
- Subtract 5 from both sides:
[tex]\[ x - 5 = 2y \][/tex]
- Divide both sides by 2 to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{x - 5}{2} \][/tex]

So, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{x - 5}{2} \][/tex]

4. Find [tex]\( f^{-1}(8) \)[/tex]
Substitute [tex]\( x = 8 \)[/tex] into the inverse function:
[tex]\[ f^{-1}(8) = \frac{8 - 5}{2} = \frac{3}{2} \][/tex]

Therefore, the value of [tex]\( f^{-1}(8) \)[/tex] is [tex]\( \frac{3}{2} \)[/tex].

So the correct answer is [tex]\( \frac{3}{2} \)[/tex].