How do you find the inverse of #f(x) = e^x - e^-x#?

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Answer 1 · 2016-01-30T01:03:16

Let #y=f(x)# and rearrange into a quadratic in #e^x# to find:

#f^(-1)(y) = ln((y+sqrt(y^2+4))/2)#

Let #y = e^x-e^-x#

Then #y(e^x) = (e^x)^2-1#

So #(e^x)^2-y(e^x)-1 = 0#

#e^x = (y+-sqrt(y^2+4))/2#

One of these roots is negative, requiring #x# to be non-Real.

So only the positive root is useful for #e^x# as a Real valued function of Real values of #x#:

#x = ln((y+sqrt(y^2+4))/2)#

That is:

#f^(-1)(y) = ln((y+sqrt(y^2+4))/2)#