Domain of a function ($\mathrm{dom}f$)

The domain of a function is the set of inputs on which it is defined.

An algebraic simplification proves equality only where both original and transformed expressions are defined. If a cancelled factor vanishes at a point, the simplified expression may instead describe an extension to that point. For example,holds only when $\cos x\ne 0$ and $\cos x+\sin x\ne 0$. The right side is zero at $x=\pi /2+k\pi$, but the left side is undefined there. A graph of the original function must retain those holes. The extension may be useful, but it is a different function with a larger domain of a function.

For a partial function $f:X\rightharpoonup Y$, the domain is the subset $\{x\in X:f(x)\text{ is defined}\}$. Outside this subset, evaluation of $f$ is undefined.