Adjusted Composition

Algebra Level 3

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

For x1x\ne-1, consider a function ff as described above for some constant α\alpha. What is the value of α \alpha for which f(f(x))f(f(x)) is an identity function?

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

Image Credit: Flickr Eliana Lúcio.
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