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]}