Answer: The required inverse function is given by
\(f^{-1}(x)=\dfrac{1}{x-1}.\)
Step-by-step explanation: We are given to find the inverse function of the following function :
\(f(x)=\dfrac{x+1}{x}~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\)
Let us consider that
f(x) = y, which gives that
\(x=f^{-1}(y).\)
From equation (i), we have
\(f(x)=\dfrac{x+1}{x}\\\\\\\Rightarrow y=\dfrac{f^{-1}(y)+1}{f^{-1}(y)}\\\\\\\Rightarrow yf^{-1}(y)=f^{-1}(y)+1\\\\\Rightarrow yf^{-1}(y)-f^{-1}(y)=1\\\\\Rightarrow (y-1)f^{-1}(y)=1\\\\\Rightarrow f^{-1}(y)=\dfrac{1}{y-1}.\)
Thus, the required inverse function is given by
\(f^{-1}(x)=\dfrac{1}{x-1}.\)
