2
$\begingroup$

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

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.

  • 0
    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

2 Answers 2

4

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

  • 0
    Not sure the OP knows about projective geometry. I was just giving a hint anyway.2019-01-28
5

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}) = ... $