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})

y = 3 cos (x - 1) - 3

f (x) = ln (x) + 3

f (- x) = \frac{x ^{3} + 3 x}{x}

y = (x + 3) (x - 1)

f (x) = \frac{2}{x ^{2} + 1}

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)