Functions, domains, and representations

Functions and Domains

A function assigns exactly one output to every input in its domain. Walk the input along the axis and $f(x)=\frac{1}{x-2}$ answers every time, until $x = 2$, where it asks for $1/0$ and has nothing to hand back.

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

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

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. Always state the domain before computing.

Representations and Symmetry

Functions take different mathematical shapes depending on how they are written.

Symmetry check: A function is even when $f(-x)=f(x)$, meaning it is symmetric about the $y$-axis. A function is odd when $f(-x)=-f(x)$, meaning it is symmetric about the origin.

A formula without its explicit domain is an incomplete model. When checking properties, always verify both the algebraic rule and the underlying domain constraints to avoid false conclusions about range and symmetry.