How to approach an equation of the form $f(x)=y$ where $f$ is recursively defined?

Question

This is a homework problem so I am not looking for someone to solve it for me. I would like to know how I should approach this problem, or in what direction I should research to figure it out myself.

For $y \in \mathbb N$, $y > 1$, find $x$ for which $f(x) = y$. Given:

• $f(0)=1$
• $f(1)=1$
• $f(2)=2$
• $f(2x)=f(x)+f(x+1)+x$
• $f(2x+1)=f(x)+f(x−1)+1$

Answer 1

Solve for $1$:

$f(2x+1) - f(x) - f(x-1) = 1 = f(1)$

Solve for $2$:

$f(2) = f(1) + f(2) + 1 = 1 + 2 + 1 = 4 \neq 2$

Contradiction