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$$f \left (\frac{x}{x+1} \right) = x^2 \implies f(x)=\;?$$

I encountered this exercise, and and don't know how to solve it.

In what category of math does this belong?

In what book/website I can study/exercise myself?

thank you very much.

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    I don't think you question does make a lot of sense: $f(x/x+1) = f(2) =x2$, from this you can say nothing about $f(x)$ apart from the obvious. Are you missing some brackets? Do you want to say $f(x/(x+1))= x^2$?2011-03-09
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    yes sorry, I mean f(x/(x+1))=x²2011-03-09
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    I'd look at a projective transformation that takes the hyperbola $y = \frac{x}{x + 1}$ to the parabola $y = x^2$. Not sure if it fits the parametrization, though.2011-03-09

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Hint: Write $f(y/(y+1))= y^2$, now solve $y/(y+1)=x$ for $y$.

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    Oh thanks, that works :D. Where can I found exercises like this? any book/site advice?2011-03-09
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    Not sure where you can find that specific type of exercises. It's a kind of exercise on making implicitly defined functions explicitly defined. So books about implicit functions I guess.2011-03-09
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    What about the points where $\frac{x}{x + 1}$ and its 'inverse' are not defined? They will be fine if we work on the projective line, but in the real case taking this inverse looks ugly.2011-03-09
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    Not sure the OP knows about projective geometry. I was just giving a hint anyway.2011-03-09
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This is an exercise about "change of variable" or "variable substitution". This is an important technique in elementary calculus and topics that use calculus. You'll encounter it in integration theory first, and again with problems involving differential equations etc.

The "trick" is to introduce an old variable x (already there in the problem statement) and a new variable, let's call it z, then we already know z as a function of x, $$ z := \frac{x}{x+1} $$ Now you have to calculate the inverse, x as a function of z: $$ x(z) = ...? $$ When you know that, you can calculate $x(z)^2$ and finally the right hand side of $$ f(z) = f(\frac{x}{x+1}) = ... $$