If #y=e^x#,then #f^-1(x)# is?

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# If y=e^x,then f^-1(x)?#

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Answer 1 · 2017-09-17T06:23:41

The answer is #=lnx#

I assumed that

#f(x)=e^x#

Let, #y=e^x#

The domain of #x# is #RR#

The range of #y# is #RR**^+#

#f(x)# and #f^-1(x)# are reflections in the line #y=x#

Then

#ln(y)=ln(e^x)#

#lny=x#

#x=lny#

Exchanging #x# and #y# in the last equation

#y=lnx#

So, the inverse is

#f^-1(x)=lnx#

Verification by performing the composition of the functions

#fof^-1(x)=f(f^-1(x))=f(lnx)=e^(lnx)=x#

graph{(y-x)(y-e^x)(x-e^y)=0 [-5.34, 8.705, -2.81, 4.21]}