Domain and Range

The domain of a function $f$ is the set of all input values $x$ for which $f(x)$ is defined. The range is the set of all output values that $f$ actually produces.

Common domain restrictions:

• Division: $f(x) = 1/x$ excludes $x = 0$.
• Even roots: $f(x) = \sqrt{x}$ requires $x \geq 0$ in the reals.
• Logarithms: $\ln(x)$ requires $x > 0$.

Finding the range is often harder than the domain — you must analyse the function's behaviour. For polynomials, calculus (derivatives, asymptotic analysis) helps determine the range; for trig functions, you exploit periodicity and bounded amplitude (e.g. $\sin x$ has range $[-1, 1]$).

In coding, "domain" / "range" become type signatures; in machine learning, they describe the input space and output space of a model.