Concept of reduction and algorithm
I think I see where your confusion arises. Hopefully ;).

Suppose you have a problem $\Pi \in NP$, and two algorithms $\mathcal{A}$ and $\mathcal{B}$ that both solve $\Pi$.

Assume that $\mathcal{A}$ uses algorithm $\mathcal{A'}$ as a sub-algorithm, where $\mathcal{A'}$ is acutally an algorithm for a $PSPACE$-complete problem.

Also assume that $\mathcal{B}$ is the same as $\mathcal{A}$, but instead of using $\mathcal{A'}$ as a sub-algorithm, it uses $\mathcal{B'}$, which is sufficient to solve $\Pi$, but not to solve the $PSPACE$-complete problem.

I think this is the situation you want to describe?

Now I think your confusion comes from assume that because a problem has two different algorithms, that the algorithms must have the same complexity - at least this is what I get from the analogy with the functions.

This isn't true, you can quite happily have an algorithm which solves a problem, but is actually far more powerful (i.e. it can solve a much bigger super-problem) and another algorithm that can only solve the smaller problem. These two algorithms don't really say anything about one another (at least without a lot more information).

To stretch the function analogy to its breaking point, the situation is more like $g(x) \leq f(x)+h(x)$ and $g(x) \leq f(x)+ q(x)$. Sort of. But not really.