A function $f$ from $A$ to $B$ is a relation in which every element of $A$ is paired with exactly one element of $B$. Two demands, both essential: every input is used (no element of $A$ left out), and each input gives one output (never two). We write $f : A \to B$ and $y = f(x)$.
The vertical line test captures this graphically: a curve is a function exactly when no vertical line meets it more than once (one $x$, one $y$).
Standard real functions you must recognise by formula and graph:
| Function | Rule | Domain | Range |
|---|---|---|---|
| Identity | $f(x) = x$ | $\mathbb{R}$ | $\mathbb{R}$ |
| Constant | $f(x) = c$ | $\mathbb{R}$ | $\{c\}$ |
| Modulus | $f(x) = |x|$ | $\mathbb{R}$ | $[0, \infty)$ |
| Signum | $f(x) = \dfrac{|x|}{x}\,(x\ne0)$ | $\mathbb{R}$ | $\{-1, 0, 1\}$ |
| Greatest integer | $f(x) = [x]$ | $\mathbb{R}$ | $\mathbb{Z}$ |
| Reciprocal | $f(x) = \dfrac{1}{x}$ | $\mathbb{R}\setminus\{0\}$ | $\mathbb{R}\setminus\{0\}$ |
Algebra of real functions: for functions with overlapping domains you may add, subtract, multiply and divide them pointwise: $(f \pm g)(x) = f(x) \pm g(x)$, $(fg)(x) = f(x)g(x)$, and $\left(\tfrac{f}{g}\right)(x) = \tfrac{f(x)}{g(x)}$ wherever $g(x) \ne 0$.
Finding the domain of a real function means listing every $x$ for which the rule gives a real value — exclude inputs that cause division by zero or an even root of a negative number. The range is the resulting set of outputs.
Deeper Insight — "exactly one output" is the entire idea, and it is what makes calculus possible: A function is a relation that has been disciplined by a single rule: one input, one output. That rule looks modest but it is the reason functions, not arbitrary relations, became the central object of mathematics — because a unique output per input is exactly what lets us speak of "the value of $f$ at $x$", differentiate it, integrate it, and invert it. The vertical line test is just this rule drawn on a graph. The standard functions in the table are worth memorising not as trivia but as a vocabulary: nearly every function you meet later is built by combining, transforming or composing these few, so knowing their shapes (the V of $|x|$, the steps of $[x]$, the three-level jump of the signum) lets you predict the behaviour of complicated expressions at a glance. Master "unique image" and the standard graphs, and the leap to Limits and Derivatives in Chapter 12 becomes a short step rather than a cliff.