Show that the functions f: R -->1 , infinity -->R , defined by : x=3^2x +1 , x= log x-1 are inverse of one another.

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$f (x) = 3^{2 x} + 1$

$Domain: (-\infin, \infin)$

$Range: (1, \infin)$

Replace $f(x)$ by $y$

$y=3^{2x}+1$

Interchange $x$ and $y$

$x=3^{2y}+1$

Solve for $y$

$3^{2y}=x-1$

$y=\dfrac{1}{2}\log_3(x-1)$

Replace $y$ by $f^{-1}(x)$

$f^{-1}(x)=\dfrac{1}{2}\log_3(x-1)$

$Domain: (1, \infin)$

$Range: (-\infin, \infin)$

The functions $f(x)=3^{2x}+1$ and $f^{-1}(x)=\dfrac{1}{2}\log_3(x-1)$ are inverse of one another.