If $f(f(x)) = x $ has at most 1 solution, then so does $f(x) = x$.

Question

Let f be defined on [0,1] and its values are between 0 and 1. If $f(f(x)) = x $ has at most 1 solution, then $f(x) = x$ has at most 1 solution.

Please, give me a hint how to prove this.

Answer 1

This is a special case of a more general fact: if $f:A\to B$ and $g:B\to C$ are functions, and $$g\circ f:A\to C$$ is injective, then $f$ is injective.

To see this, suppose that $f(x) = f(y)$ for some $x, y\in A$. Then $gf(x) = gf(y)$, so $x=y$.

Answer 2

show that every solution of $f(x) = x$ is also a solution of $f(f(x)) = x$ which is in fact rather obvious.

Answer 3

Hint: Try contradiction. Suppose $f(x)=x$ has two solutions - what happens then?

Answer 4

ANSWER:

Suppose $f(x) = x$ and $f(x') = x'$. Then $f(f(x)) = f(x) = x$ and $f(f(x')) = f(x') = x'$. By assumption that there is at most one solution to $f(f(x)) = x$, $x = x'$. This proves there is at most one solution to $f(x) = x$.