Question

if $f(x)$ and $f^{-1}(x)$ are inverse functions of each other and $f(x) = 2x + 5$, what is $f^{-1}(8)$? options: $3\frac{3}{2}$, $23$, $\frac{41}{8}$, $-1$

Explanation:

Step1: Recall the property of inverse functions

If \( f(x) \) and \( f^{-1}(x) \) are inverse functions, then \( f(f^{-1}(y)) = y \) and \( f^{-1}(f(x)) = x \). Also, to find \( f^{-1}(8) \), we can set \( f(x)=8 \) and solve for \( x \), because if \( f(a) = b \), then \( f^{-1}(b)=a \).

Step2: Set up the equation

Given \( f(x)=2x + 5 \), set \( f(x)=8 \):
\[
2x+5 = 8
\]

Step3: Solve for \( x \)

Subtract 5 from both sides:
\[
2x=8 - 5=3
\]
Divide both sides by 2:
\[
x=\frac{3}{2}
\]

Answer:

\(\frac{3}{2}\)