Question

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

Answer 1

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

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

$Range: (1, \infin)$

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.