Question

1. the inverse of $f(x) = 3^x + 4$ is:

a. $f^{-1}(x) = \log_3(x - 4)$
b. $f^{-1}(x) = \log_4(x - 3)$
c. $f^{-1}(x) = 4^x + 3$
d. $f^{-1}(x) = -3^x - 4$

Explanation:

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

We start with the function \( f(x) = 3^x + 4 \). Replace \( f(x) \) with \( y \), so we have \( y = 3^x + 4 \).

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

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x = 3^y + 4 \).

Step3: Solve for \( y \)

First, subtract 4 from both sides of the equation: \( x - 4 = 3^y \).
Then, to solve for \( y \) when the equation is in the form \( a = b^c \), we use the logarithmic form \( \log_b(a)=c \). Here, \( b = 3 \), \( a = x - 4 \), and \( c = y \). So we get \( y=\log_3(x - 4) \).

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

Since \( y \) represents the inverse function, we replace \( y \) with \( f^{-1}(x) \). So \( f^{-1}(x)=\log_3(x - 4) \).

Answer:

A. \( f^{-1}(x)=\log_{3}(x - 4) \)