What is the inverse of f(x)=2e^x-e^{-x} ?

The inverse of f(x)=2e^x-e^{-x} is ln((x+sqrt(x^2+8))/4),ln((x-sqrt(x^2+8))/4)

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inverse of f(x)=2e^x-e^{-x}

inverse

f (x) = 2 e^{x} - e^{- x}

ln (\frac{x + x ^{2} + 8}{4}), ln (\frac{x - x ^{2} + 8}{4})

Solution steps

y = 2 e^{x} - e^{- x}

Replace

x

with

y

x = 2 e^{y} - e^{- y}

Solve for

y, x = 2 e^{y} - e^{- y}

ln (\frac{x + x ^{2} + 8}{4}), ln (\frac{x - x ^{2} + 8}{4})

12 x (x - 4)

f (x) = e^{- x}

y = - \frac{2}{3} x, (4, - 8)

f (x) = \frac{3 x - 24}{2 x ^{2} - 8 x - 64}

\frac{k + ln ( x )}{x}

What is the inverse of f(x)=2e^x-e^{-x} ?
The inverse of f(x)=2e^x-e^{-x} is ln((x+sqrt(x^2+8))/4),ln((x-sqrt(x^2+8))/4)