Question

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

Answer 1

Solution:

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

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

$Range: (1, \infin)$

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.

Hence, proved.