f= ln(2x-1) Find the inverse function stating its domain.
1. You have:
$\displaystyle f: y = \ln(2x-1)$ , domain $\displaystyle D=\left(\frac12\ ,\ \infty\right) $ and range $\displaystyle R=\mathbb{R}$
2. Then the inverse function is:
$\displaystyle f^{-1}:x=\ln(2y-1)\ ,\ D_{f^{-1}}=\mathbb{R}\ and\ R_{f^{-1}}= \left(\frac12\ ,\ \infty\right)$
That means:
$\displaystyle x=\ln(2y-1)~\implies~e^x=2y-1~\implies~\boxed{y=\dfrac{e^x+1}{2}}$