# How do you find the inverse of #y=ln(x/(x-1))#?

To find the inverse of ( y = \ln\left(\frac{x}{x-1}\right) ):

- Swap the roles of ( x ) and ( y ), making ( y ) the independent variable.
- Solve the resulting equation for ( x ).
- The resulting expression represents the inverse function.

After swapping ( x ) and ( y ), the equation becomes: [ x = \ln\left(\frac{y}{y-1}\right) ]

To solve for ( x ), exponentiate both sides with base ( e ): [ e^x = \frac{y}{y-1} ]

Solve for ( y ): [ y(e^x - 1) = e^x ] [ y = \frac{e^x}{e^x - 1} ]

Thus, the inverse function is: [ f^{-1}(x) = \frac{e^x}{e^x - 1} ]

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First of all, let's establish your domain:

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Now let's start with finding the inverse.

To procede, take the reciprocal on both sides:

Finally, take the reciprocal on both sides again:

To avoid double fractions, this can be rephrased in:

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Thus, your inverse function is

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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