What is the inverse of f(x)=e^x-e^{-x} ?

The inverse of f(x)=e^x-e^{-x} is ln((x+sqrt(x^2+4))/2),ln((x-sqrt(x^2+4))/2)

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inverse of f(x)=e^x-e^{-x}

inverse

f (x) = e^{x} - e^{- x}

ln (\frac{x + x ^{2} + 4}{2}), ln (\frac{x - x ^{2} + 4}{2})

Solution steps

y = e^{x} - e^{- x}

Replace

x

with

y

x = e^{y} - e^{- y}

Solve for

y, x = e^{y} - e^{- y}

ln (\frac{x + x ^{2} + 4}{2}), ln (\frac{x - x ^{2} + 4}{2})

What is the inverse of f(x)=e^x-e^{-x} ?
The inverse of f(x)=e^x-e^{-x} is ln((x+sqrt(x^2+4))/2),ln((x-sqrt(x^2+4))/2)