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

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

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

inverse

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

ln (\frac{x}{x - 1})

Solution steps

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

Replace

x

with

y

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

Solve for

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

ln (\frac{x}{x - 1})

y = (x - 5) 2 - 4

y = 5 - x

f (x) = 1 0^{x} + 4

f (x) = - 4 x^{2} - 12 x - 2

f (x) = \frac{6 + x}{7}

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