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.

The Answer to the Question
is below this banner.

Can't find a solution anywhere?

NEED A FAST ANSWER TO ANY QUESTION OR ASSIGNMENT?

You will get a detailed answer to your question or assignment in the shortest time possible.

Here's the Solution to this Question

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.