What is the inverse of f(x)=2e^x-1/(e^x) ?

The inverse of f(x)=2e^x-1/(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-1/(e^x)

inverse

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

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

Solution steps

y = 2 e^{x} - \frac{1}{e ^{x}}

Replace

x

with

y

x = 2 e^{y} - \frac{1}{e ^{y}}

Solve for

y, x = 2 e^{y} - \frac{1}{e ^{y}}

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

- \frac{3}{2} cos (3 x - \frac{1}{2}) + 2

\frac{x - 1}{2 x + 1}

3 sin (x)

f (y) = 2 x + b

What is the inverse of f(x)=2e^x-1/(e^x) ?
The inverse of f(x)=2e^x-1/(e^x) is ln((x+sqrt(x^2+8))/4),ln((x-sqrt(x^2+8))/4)