# Calculate the inverse of $s(x) = \frac{1+f(x)}{1-f(x)}$ in terms of $f^{-1}$

Calculate the inverse of $s(x) = \frac{1+f(x)}{1-f(x)}$ in terms of $f^{-1}$,f is a $1-1$ function with inverse $f^{-1}$

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Can you say why you're interested in this question? – Jonah Sinick Oct 29 '12 at 6:57

Assumption: $s^{-1}$ function exists, then,

Let $$y=s(x)=\frac{1+f(x)}{1-f(x)}\implies y-yf(x)=1+f(x)\implies f(x)=\frac{y-1}{y+1}\implies x=f^{-1}(\frac{y-1}{y+1})\implies s^{-1}(y)=f^{-1}(\frac{y-1}{y+1})$$ or $$s^{-1}(x)=f^{-1}(\frac{x-1}{x+1})$$

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Thank you soo much! You're a life saver. – Jack Oct 29 '12 at 5:49
@Jack: Whenevwer you have $y=f(x)$ then you can write $x=f^{-1}(y)$. – S. Snape Oct 29 '12 at 6:52
@Jack: If $f^{-1}$ exists then if $y=f(x)\implies$ $x=f^{-1}(y)$ – Aang Oct 29 '12 at 10:42
A thought: IF we consider the function $$g(x) = \frac{1 + x}{1 - x}$$ Then obviusly, $s = g \circ f$. Is $g$ injective??? What do you know about composition of injective functions? Assume $x \neq 1$.