Question

find $f(x)$ given that $f^{-1}(x) = \log_{9}(x)$. $f(x) = $

Explanation:

Step1: Recall inverse function relationship

If \( y = f^{-1}(x) \), then \( x = f(y) \). Given \( f^{-1}(x)=\log_{9}(x) \), let \( y = \log_{9}(x) \).

Step2: Convert logarithmic to exponential form

By the definition of logarithms, if \( y=\log_{b}(a) \), then \( a = b^{y} \). Here, \( b = 9 \), \( y=\log_{9}(x) \) implies \( x = 9^{y} \).

Step3: Find \( f(x) \)

Since \( x = f(y) \) when \( y = f^{-1}(x) \), and we have \( x = 9^{y} \), we can replace \( y \) with \( x \) to get \( f(x)=9^{x} \).

Answer:

\( 9^{x} \)