problem: Let $f(x) = x - 3$ and $q(x) = bx +1$.  If $f(q(1)) = -3$, what is $b$?
solution: We have $q(1) = b\cdot 1 + 1 = b+1$, so $f(q(1)) = f(b+1)$. Applying the definition of $f$, we have $f(q(1)) = f(b+1) = (b+1) - 3 = b-2$.  Therefore, the equation $f(q(1)) = -3$ gives us $b-2 = -3$, so $ b= \boxed{-1}$.