To find the inverse function [tex]\( f^{-1}(x) \)[/tex] for a given function [tex]\( f(x) \)[/tex], you need to follow these steps:
1. Start with the function [tex]\( f(x) = 2x + 5 \)[/tex].
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex] to get [tex]\( y = 2x + 5 \)[/tex].
3. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse: [tex]\( x = 2y + 5 \)[/tex].
4. Solve for [tex]\( y \)[/tex] to express it in terms of [tex]\( x \)[/tex]:
[tex]\[
x = 2y + 5
\][/tex]
Subtract 5 from both sides:
[tex]\[
x - 5 = 2y
\][/tex]
Divide both sides by 2:
[tex]\[
y = \frac{x - 5}{2}
\][/tex]
Therefore, the inverse function is [tex]\( f^{-1}(x) = \frac{x - 5}{2} \)[/tex].
5. Now, we need to find [tex]\( f^{-1}(8) \)[/tex]. Substitute [tex]\( x = 8 \)[/tex] into the inverse function:
[tex]\[
f^{-1}(8) = \frac{8 - 5}{2}
\][/tex]
6. Perform the arithmetic:
[tex]\[
f^{-1}(8) = \frac{3}{2} = 1.5
\][/tex]
Thus, [tex]\( f^{-1}(8) = 1.5 \)[/tex]. Among the given options, [tex]\( \frac{3}{2} \)[/tex] is equivalent to 1.5. Therefore, the correct answer is:
[tex]\(\boxed{\frac{3}{2}}\)[/tex]
