To find the inverse of the logarithmic function [tex]\( f(x) = \log_9(x) \)[/tex], we need to follow the definition and properties of logarithms and exponents.
1. Understand the Function [tex]\( f(x) = \log_9(x) \)[/tex]:
- The function [tex]\( f(x) = \log_9(x) \)[/tex] means that [tex]\( f(x) \)[/tex] is the power to which the base 9 must be raised to obtain [tex]\( x \)[/tex].
- In mathematical terms, if [tex]\( y = \log_9(x) \)[/tex], then [tex]\( 9^y = x \)[/tex].
2. Define the Inverse Function [tex]\( f^{-1}(x) \)[/tex]:
- The inverse function [tex]\( f^{-1}(x) \)[/tex] should reverse the original function [tex]\( f(x) \)[/tex]. This means if [tex]\( y = f(x) \)[/tex], then [tex]\( x = f^{-1}(y) \)[/tex].
- Given [tex]\( y = \log_9(x) \)[/tex], we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
3. Express [tex]\( x \)[/tex] in Terms of [tex]\( y \)[/tex]:
- From the equation [tex]\( y = \log_9(x) \)[/tex], we rewrite it as [tex]\( 9^y = x \)[/tex].
4. Identify the Inverse Function:
- The equation [tex]\( 9^y = x \)[/tex] tells us that [tex]\( f^{-1}(y) = 9^y \)[/tex]. Since [tex]\( y \)[/tex] is the argument of the inverse function, we can generalize this as [tex]\( f^{-1}(x) = 9^x \)[/tex].
5. Verification with the Options Provided:
- [tex]\( f^{-1}(x) = x^9 \)[/tex]
- This would suggest that the original function [tex]\( f(x) \)[/tex] would be [tex]\( \sqrt[9]{x} \)[/tex], not [tex]\( \log_9(x) \)[/tex].
- [tex]\( f^{-1}(x) = -\log_9(x) \)[/tex]
- This is not the correct inverse because it does not reverse the logarithm function.
- [tex]\( f^{-1}(x) = 9^x \)[/tex]
- This correctly reverses [tex]\( \log_9(x) \)[/tex] since [tex]\( 9^{(\log_9(x))} = x \)[/tex].
- [tex]\( f^{-1}(x) = \frac{1}{\log_9(x)} \)[/tex]
- This is also incorrect because it does not reverse the logarithm function.
Therefore, the correct inverse function of [tex]\( f(x) = \log_9(x) \)[/tex] is [tex]\( f^{-1}(x) = 9^x \)[/tex].
Thus, the correct option is:
[tex]\[ f^{-1}(x) = 9^x \][/tex]
So, the answer is:
[tex]\[ \boxed{3} \][/tex]
