Question

(unit 4) what is the inverse of the function $f(x) = 3^{x - 2}$?
(a)
a $log_3(x) + 2$
b $log_3(x) - 2$
c $3^{x + 2}$
d $-3^{x + 2}$

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start with the function \( f(x) = 3^{x - 2} \). Replace \( f(x) \) with \( y \), so we have \( y = 3^{x - 2} \).

Step2: Swap \( x \) and \( y \)

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x = 3^{y - 2} \).

Step3: Solve for \( y \) using logarithms

We know that if \( a^b = c \), then \( \log_a(c) = b \). Applying the logarithm with base 3 to both sides of \( x = 3^{y - 2} \), we get \( \log_3(x) = y - 2 \).
Then, solve for \( y \) by adding 2 to both sides: \( y = \log_3(x) + 2 \).

Step4: Replace \( y \) with \( f^{-1}(x) \)

The inverse function is \( f^{-1}(x) = \log_3(x) + 2 \), which corresponds to option A.

Answer:

A. \( \log_3(x) + 2 \)