Mathematics

Problem

Let

f:\mathbb R\to(0,\infty)

be given by

f(x)=e^{kx}

and

g:(0,\infty)\to\mathbb R

be given by

g(y)=\ln(py)

, where

k>0

and

p>0

. For which ordered pair

(k,p)

does each function serve as the inverse of the other, that is,

g\circ f=I_{\mathbb R}

and

f\circ g=I_{(0,\infty)}

? a)

(k=2,\;p=1)

(k=1,\;p=1)

(k=1,\;p=e)

(k=2,\;p=\dfrac12)