functional equations - If a function is like $f(f(y))=a^2+y$, does it imply that $f$ is surjective?

If a function is like $f(f(y))=a^2+y$, does it imply that $f$ is surjective?

Just for an example, consider this:

Find all functions $f:\mathbb{R}\mapsto \mathbb{R}$ such that

$$f(xf(x)+f(y))=(f(x))^2+y$$ for all real values of $x,y$.
It's solution begins as follows:
Let $f(0)=a$. Setting $x=0$ we get $$f(f(y))=a^2+y ~ \forall y\in \mathbb{R}$$

Now we can say that the range of $a^2+y$ is all real numbers, so $f$ is surjective.

What if $f\in (a,b)$ where $(a,b)$ is some smaller interval?