Option (A) Worst case of algorithm is $O(g(n))$, means that in worst case the algorithm's time complexity grows (asymptotically or for sufficiently large $n$) no faster than $g(n)$.
Best case of algorithm is $\Omega(g(n)$, means that in best case inputs, the algorithm's time complexity grows at least as fast as $g(n)$.
From these two statements we can conclude that best case time complexity of algorithm is $\Theta(g(n))$ and worst case time complexity of algorithm is $\Theta(g(n))$.
From this we can conclude that, average case time complexity will also be $\Theta(g(n))$.
Which implies algorithm's running time complexity is $\Theta(g(n))$ which in turn implies algorithm's running time complexity is $O(g(n))$.
That is, Option (A) implies running time complexity is $O(g(n))$.
But running time complexity is $O(g(n))$ does not imply Option (A).
For this counter example is Insertion sort, it's running time complexity is $O(n^2)$, worst case time complexity is $O(n^2)$ but it's best case time complexity is not $\Omega(n^2)$.
Therefore, option (A) must be False.