If $B$ is worse than $A$ on some inputs, how do their worst-case time complexities compare? You are given two algorithms $A$ and $B$, with worst-case time complexity $f_A(n)$ and $f_B(n)$, respectively.

Assume:

(i) For each $n$ there exists an input $x$ of size $n$ such that the number of steps of algorithm $A$ on $x$ is half the number of steps of $B$ on $x$.

(ii) $f_A(n) = \Omega(h(n))$ for some positive function $h(n)$.


Is it possible that $f_B(n) = \Omega(h(n))$
Is it necessary that $f_B(n) = \Omega(h(n))$?


Explain your answers.
You are essentially asking the following question:


  Suppose that $f(n) \geq g(n)$ for some $n$. Is there a universal constant $C$ such that $\min_n f(n) \geq C \min_n g(n)$?


Hopefully this formulation makes it easier to answer.