#math/calculus Let $D\subseteq\mathbb{R}$. A **function** $f$ with domain $D$ assigns exactly one real number $f(x)$ to each $x\in D$. Its graph is the set $\{(x,f(x)):x\in D\}$, and its range is the set of output values. A set of points is the graph of a function exactly when every vertical line intersects it at most once—the **vertical-line test**. The circle $x^{2}+y^{2}=1$ fails this test because most $x\in(-1,1)$ correspond to two values of $y$. Its upper and lower semicircles are functions: $ y=\sqrt{1-x^{2}} \qquad\text{and}\qquad y=-\sqrt{1-x^{2}}, \qquad -1\leq x\leq 1. $ The formula and domain are both part of a function. For example, $f(x)=3/(5-x)$ is a function on any domain that excludes $x=5$. For graph transformations, $f(x-p)+q$ shifts the graph right by $p$ and up by $q$; $af(x)$ scales it vertically and reflects it across the $x$-axis when $a<0$. See [[0R5 - Circles and Parabolas]]. > [!Practice] > Marsden & Weinstein, p. 48, exercises 83–84.