Find all functions $f(x)$ where $2f(x)+1=f(2x+1)$ for all real $x$

Question

I have not gotten much except $f(-1)=-1$ from subbing $x=-1$.

If $f$ is linear I get $f(x)=(c+1)x+c$ from subbing $f(x)=ax+b$.

I'm not really sure how to show that $f$ is linear, or narrow it down any further.