# Does FPT allow for doubling the parameter?

I have recently come across a result that showed that a given problem is in FPT when parameterized by the treewidth of a graph. However, they did this by showing that the problem is in FPT when parameterized by the treewidth of an augmented version of the graph that had treewidth of at most 2k+1 when the treewidth of the original graph is k.

Why is it possible do this?

Let $$k$$ be the parameter. Is an algorithm has a running time of the form $$O(f(2k+1) \cdot \text{poly}(n))$$ for a suitable function $$f$$, then it also has a running time of $$O(g(k) \cdot \text{poly}(n) )$$ for $$g(k) = f(2k+1)$$.