What is the inverse of #y= e^(x-1)-1# ?

1 Answer
Lotusbluete
Dec 7, 2015

#f^(-1)(x) = ln(x+1) +1#

Explanation:

To compute the inverse, you need to follow the following steps:

1) swap #y# and #x# in your equation:

#x = e^(y-1) - 1#

2) solve the equation for #y#:

... add #1# on both sides of the equation...

#x + 1 = e^(y-1) #

... remember that #ln x# is the inverse function for #e^x# which means that both #ln(e^x) = x# and #e^(ln x) = x# hold.
This means that you can apply #ln()# on both sides of the equation to "get rid" of the exponential function:

#ln(x+1) = ln(e^(y-1))#

#ln(x+1) = y-1#

... add #1# on both sides of the equation again...

#ln(x+1) + 1 = y#

3) Now, just replace #y# with #f^(-1)(x)# and you have the result!

So, for

#f(x) = e^(x-1) - 1#,

the inverse function is

#f^(-1)(x) = ln(x+1) +1#

Hope that this helped!

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