CIIM 2021 Problem 4

Let $\mathbb{Z}^{+}$ be the set of positive integers.a) Prove that there is only one function $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ , strictly increasing, such that $f(f(n))=2n+1$ for every $n\in \mathbb{Z}^{+}$ .b) For the function in a. Prove that for every $n\in \mathbb{Z}^{+}$ $\frac{4n+1}{3}\leq f(n)\leq \frac{3n+1}{2}$ c) Prove that in each inequality side of b the equality can reach by infinite positive integers $n$ .