How !x operator works?!

Answer 1

Allow me to do a little correction to your query:

From what I perceive, 'x' may be a boolean variable which holds value of either true, or false,

The operator in question here is the ! operator.
The ! operator is the NOT operator, which is used to return the negation of the truth value of whatever comes after it.

e.g.

bool x = true;
cout << x << endl;
cout << !x << endl;

// outputs 1 and 0.

Answer 2

! is a unary operator that complements a binary digit or a boolean. When x is an integer, it is converted to 1 (true) if x is nonzero otherwise it is converted to 0 (false). Now you apply the ! operator. !true == false and ! false==true