Given that
We have to find the inverse of a function and the function is
[tex]g(x)=2^{x-1}[/tex]Explanation -
The steps to find the solution are -
(1). First, we will consider the function equal to y as
[tex]\begin{gathered} g(x)=y \\ x=g^{-1}(y)----------(i) \\ y=2^{x-1} \end{gathered}[/tex](2). Now from this, we will find the value of x in terms of y.
And we will take the base of the log as 2 because the x is in the power of 2. As an exponential function.
As
[tex]\begin{gathered} y=\frac{2^x}{2} \\ Taking\text{ the }\log_2\text{ on both sides we have} \\ \log_2y=\log_2(\frac{2^x}{2}) \\ \log_2y=\log_22^x-\log_22 \\ \\ Using\text{ the formulae of log,} \\ \begin{equation*} \log_a(\frac{b}{c})=\log_ab-\log_ac \end{equation*} \\ \log_aa^p=p\times\log_aa \\ and\text{ }\log_aa=1 \\ \\ Now,\text{ } \\ \log_2y=\log_22^x-1 \\ \log_2y=x\log_22-1 \\ \log_2y=x-1 \\ x=\log_2y+1 \end{gathered}[/tex](3). Now from eq (i) we have
[tex]g^{-1}(y)=\log_2y+1[/tex](4). At last, we will replace y with x. Then,
[tex]g^{-1}(x)=\log_2x+1[/tex]Final answer -
The final answer is [tex]g^{-1}(x)=\log_2x+1[/tex]