Question

find the inverse of the function $f(x) = \log_{8}(x - 1)$.
a $f^{-1}(x) = 8^{x + 1}$
b $f^{-1}(x) = 8^{x} + 1$
c $f^{-1}(x) = 8^{x}$
d $f^{-1}(x) = x^{8 - 1}$

Explanation:

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

We start by letting \( y = \log_{8}(x - 1) \). This is a standard first step when finding the inverse of a function, as it helps us work with the equation more easily.

Step2: Convert logarithmic to exponential form

Recall that if \( \log_{a}(b)=c \), then \( a^{c}=b \). Applying this to \( y = \log_{8}(x - 1) \), we get \( 8^{y}=x - 1 \). The logarithm \( \log_{a}(b) \) and the exponential function \( a^{x} \) are inverses of each other, so we can use this property to convert between the two forms.

Step3: Solve for \( x \)

To solve for \( x \), we add 1 to both sides of the equation \( 8^{y}=x - 1 \). This gives us \( x = 8^{y}+1 \).

Step4: Swap \( x \) and \( y \)

After solving for \( x \) in terms of \( y \), we swap \( x \) and \( y \) to get the inverse function. So we replace \( y \) with \( f^{-1}(x) \) and \( x \) with \( y \), resulting in \( f^{-1}(x)=8^{x}+1 \).

Answer:

B. \( f^{-1}(x) = 8^{x}+1 \)