How do you define a function to get the output you need for a particular parameter?

Question

Suppose we have a function $y(x)$ such that

$y (\frac{-e^{-2\lambda} + e^{-\lambda}}{1-e^{-2\lambda}}) = \lambda$

How can I determine $y(x)$? Are there steps that outline how to solve such a problem? If so, could somebody point me to them?

For example, an easy scenario is find $g(x)$ such that $g(\frac{1}{\lambda})=\lambda$. Obviously $g(x)=\frac{1}{x}$. But this is easy and I didn't have to do much to realize what $g(x)$ is. However, in the problem above it's more complicated.

Answer 1

$$y(x)=\log\left(\frac1x-1\right)\qquad x\in(0,1)$$

Answer 2

Simplify the argument to obtain 1/(exp(lambda)+1)

So you need a map from the previous number to lambda

Or you need an inverse function for 1/(exp(lambda)+1))

Switch y and x to obtain... F(x) could be ln(1/x-1) lambda is zero, x is is not well defined but F(x) =0 As a limit this makes sense