To find [tex]\( f^{-1}(8) \)[/tex], we first need to determine the inverse function [tex]\( f^{-1}(x) \)[/tex] of the given function [tex]\( f(x) = 2x + 5 \)[/tex].
Here are the detailed steps to find the inverse function:
1. Set up the equation:
We start with the function [tex]\( f(x) = 2x + 5 \)[/tex] and replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = 2x + 5
\][/tex]
2. Swap variables:
To find the inverse, we switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = 2y + 5
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
To isolate [tex]\( y \)[/tex], we follow these steps:
[tex]\[
x - 5 = 2y
\][/tex]
[tex]\[
y = \frac{x - 5}{2}
\][/tex]
Therefore, the inverse function is:
[tex]\[
f^{-1}(x) = \frac{x - 5}{2}
\][/tex]
4. Evaluate [tex]\( f^{-1}(8) \)[/tex]:
We substitute [tex]\( x = 8 \)[/tex] into the inverse function:
[tex]\[
f^{-1}(8) = \frac{8 - 5}{2}
\][/tex]
[tex]\[
f^{-1}(8) = \frac{3}{2}
\][/tex]
So, [tex]\( f^{-1}(8) = \frac{3}{2} \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{\frac{3}{2}}
\][/tex]
