Why does F + F' = 1?
The point is, it really doesn't matter what the function \$f()\$ actually is. The key fact is that its output is a single binary value.

It is a fundamental fact in Boolean algebra that the complement of a binary value is true whenever the value itself is false. This is known as the law of excluded middle. So ORing a value with its complement is always true, and ANDing a value with its complement is always false.

It's nice that you were able to derive the specific function \$f'()\$, but that's actually irrelevant to the actual question!