Find the inverse of the function.
-$$f ( x ) = e ^ { - x } + 2$$
A) $f ^ { - 1 } ( x ) = - e ^ { x } - 2$
B) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \mathrm { e } ^ { \mathrm { x } } - \frac { 1 } { \mathrm {~s} }$
C) $f ^ { - 1 } ( x ) = - \ln ( x - 2 )$
D) $f ^ { - 1 } ( x ) = \ln 2 - \ln x$

Question

Quizplus · Accepted Answer

To find the inverse, we swap $x$ and $y$, then solve for $y$. Starting with $y = e^{-x} + 2$, we swap to get $x = e^{-y} + 2$. Solving for $y$, we subtract 2 from both sides to get $x - 2 = e^{-y}$, then take the natural logarithm of both sides to isolate $y$, resulting in $y = -\ln(x - 2)$.

Find the Inverse of the Function f(x)=e−x+2f ( x ) = e ^ { - x } + 2f(x)=e−x+2

Find the Inverse of the Function $f ( x ) = e ^ { - x } + 2$