If $Q$ reduces to $L$ then $\overline{Q}$ reduces to $\overline{L}$ The following exercise is taken from Chapter 17 of Languages and Machines by Thomas Sudkamp:


  Let $Q$ be a language reducible to a language $L$ in polynomial time. Prove that $\overline{Q}$ is reducible to $\overline{L}$ in polynomial time.


I am not sure if my solution idea is correct. Anyone knows the correct solution of this exercise?

My idea is: I use the definition of NP-complete. Q is in NP class and L is NP-complete. The function f to reducible in polynomial time Q to L, is also the function f to reducible in polynomial time notQ to notL
By hypothesis $Q \le_p L$, i.e., there exists a poly-time computable function $f(x)$ such that $x \in L \iff f(x) \in Q$. Then: 
$$
x \in \overline{L} \iff x \not\in L \iff f(x) \not\in Q \iff f(x) \in \overline{Q}.
$$

Therefore $\overline{L} \le_p \overline{Q}$.