Time complexity analysis of 2 arbitrary algorithms – prove or disprove


We are given 2 algorithms A and B such that for each input size, algorithm A performs half the number of steps algorithm B performs on the same input size.

We denote the worst time complexity of each one by $ g_A(n),g_B(n)$

Also, we know there’s a positive function $ f(n)$ such that $ g_A(n)\in\Omega(f(n))$

Is it possible that $ g_B(n)\in\Omega(f(n))$ ? Is it necessary?

It seems naive to think that it’s necessary, but I can’t figure out to contradict it.