Given That $f ( x ) = e ^ { x - 1 } + 3$

Multiple Choice

Given that $f ( x ) = e ^ { x - 1 } + 3$ , find $f ^ { - 1 } ( x )$ and give the domain and range of $f ^ { - 1 } ( x )$ .

A) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \ln ( \mathrm { x } - 1 ) + 3$ , domain $= ( - \infty , \infty )$ , range $= ( - \infty , \infty )$
B) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \ln ( x - 1 ) + 3$ , domain $= ( 3 , \infty )$ , range $= ( - \infty , \infty )$
C) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \ln ( x - 3 ) + 1$ , domain $= ( 0 , \infty )$ , range $= ( 0 , \infty )$
D) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \ln ( \mathrm { x } - 3 ) + 1$ , domain $= ( 3 , \infty )$ , range $= ( - \infty , \infty )$