Q.Find the domain of $\dfrac{1}{1-2\sin x}$.

Before defining relations and functions, two supporting ideas are pinned down.

Constants and variables. A constant stays fixed through a process; a variable changes. A variable is independent if free to take any value, dependent if determined by other variables -- but the labels are relative: in $x+y=1$, either $x$ or $y$ may be chosen as the independent one, with the other becoming dependent. In $A=\tfrac12bh$, $\tfrac12$ is constant, $b,h$ are independent, $A$ is dependent.

Intervals. A subset $I\subseteq R$ is an interval if it has at least two elements and contains every real number strictly between any two of its elements (no gaps). Geometrically, intervals are exactly the rays and line segments of the real line; $N,\ Z$, the evens, the odds, the primes are not intervals (they skip values). A finite interval has two finite endpoints (though still infinitely many points inside); an infinite interval extends to $\pm\infty$ (not numbers, just "unbounded" markers). Closed = contains both endpoints; open = contains neither. …