We have $g(4) = \sqrt{2(4) + 1} = \sqrt{9} = 3$, so $f(g(4)) = f(3) = 3^2 -2(3) +1 = 4$.  Since $f(3) = 4$, we have $g(f(3)) = g(4) = \sqrt{2(4) + 1} = 3$.  Therefore, $f(g(4)) -g(f(3)) = 4-3 = \boxed{1}$.


Notice that $f(g(4)) = 4$ and $g(f(3)) = 3$.  Is that a coincidence?
If $f(x) = x^2-2x+1$ and $g(x) = \sqrt{2x+1}$ what is the value of $f(g(4)) - g(f(3))$ ?
{'answer': '1', 'subject': 'Algebra', 'level': 2, 'unique_id': 'test/algebra/771.json'}