Adjusted Composition

Algebra

$\large f(x) = \frac{\alpha x}{x+1}$

For $x\ne-1$ , consider a function $f$ as described above for some constant $\alpha$ . What is the value of $\alpha$ for which $f(f(x))$ is an identity function?

Clarification: $g(x)$ is an identity function if $g(x) = x$ .

Adjusted Composition

Image Credit: Flickr Eliana Lúcio.