Given that $f ( x ) = 5 \ln 2 x$, find $f ^ { - 1 } ( x )$ and give the domain and range of $f ^ { - 1 } ( x )$.
A) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { x } - 5 }$, domain $= ( 0 , \infty )$, range $= ( - \infty , \infty )$
B) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { x } / 5 }$, domain $= ( - \infty , \infty )$, range $= ( 0 , \infty )$
C) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { x } / 5 }$, domain $= ( 0 , \infty )$, range $= ( - \infty , \infty )$
D) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \mathrm { e } ^ { \mathrm { x } - 5 }$, domain $= ( - \infty , \infty )$, range $= ( 0 , \infty )$

Question

Quizplus · Accepted Answer

The inverse function is found by solving $y = 5 \ln 2x$ for $x$, giving $x = \frac{1}{2} e^{y/5}$. The domain of $f^{-1}(x)$ is $(-\infty, \infty)$ and the range is $(0, \infty)$.

Given That f(x)=5ln⁡2xf ( x ) = 5 \ln 2 xf(x)=5ln2x

Given That $f ( x ) = 5 \ln 2 x$