Functions, domains, and representations

A function assigns exactly one output to every input in its domain.

$$f : D \subseteq \mathbb{R} \;\longrightarrow\; \mathbb{R}$$

Key ideas

• The domain $D$ collects every $x$ for which $f(x)$ is defined. For $f(x)=\dfrac{1}{x-2}$ we need $x \neq 2$, so $D = \mathbb{R}\setminus\{2\}$.
• The range (or image) is the set of all values $f$ actually takes.
• The number sets nest: $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$.

Common representations

Symmetry check — A function is even when $f(-x)=f(x)$ (symmetric about the $y$-axis) and odd when $f(-x)=-f(x)$ (symmetric about the origin).

Tip: Always state the domain before computing. A formula without its domain is an incomplete model.

Common pitfall: A formula without its domain is only half a function. $f(x) = 1/x$ and "$1/x$ for $x > 0$" are different mathematical objects with different properties.