surjectivity and inverse of a function How to show that $f(x)=x^2/(1+x^2)$ is surjective on the codomain $[0,1)$ from $(0,\infty)$?
What would be its inverse? I already proved injectivity, so it must be bijective if $f(x)$ is also surjective.
I get stuck with recursive definition: $f^{-1} (x) = \sqrt{x+xf^{-1}(x)^2}$.

Observe first that $f(x) = 0 \Leftrightarrow x = 0$ and then your function is not surjective with that domain.
You can prove easily that $f$ is increasing and therefore as $f(0) = 0$ and $\lim_{x\to \infty} f(x) = 1$ then you have that $f$ is surjective.
To find its inverse you have to isolate $x$ form $f(x) = x^2/(1+x^2)$.
