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.