Functions and Their Graphs

Functions are the language of calculus. In this lesson you'll pin down exactly what a function is, evaluate it with function notation, find where it is defined, and learn to build new functions by combining and composing old ones — then classify functions by their symmetry and handle piecewise, absolute-value, and step functions.

A relation is any set of ordered pairs $(x, y)$. A function is a special relation: it pairs each input with exactly one output.

So $\{(1, 2), (2, 3), (4, 7)\}$ is a function — each input appears once — while $\{(1, 2), (1, 4), (2, 6)\}$ is not, because the input $1$ is paired with both $2$ and $4$.

Function notation names the rule and its input at once. Writing $f(x) = x^2 + 2x - 3$ means “the function $f$ takes an input $x$ and returns $x^2 + 2x - 3$.” The symbol $x$ is just a placeholder — you can substitute a number or an entire expression.

Substituting an expression powers the difference quotient — the average rate of change of $f$ over a step of width $\Delta x$. It is the algebraic engine behind the derivative you'll meet later.

The domain is the set of all inputs for which the function is defined. Unless a domain is stated, assume the implied domain: all real numbers that produce a real output. Two situations create restrictions:

• Even roots: the radicand must be $\ge 0$. For $\sqrt{x - 3}$ require $x - 3 \ge 0$, so the domain is $[3, \infty)$.
• Denominators: never zero. For $\dfrac{1}{x - 2}$ exclude $x = 2$.

Absolute values and odd roots accept every real input to those operations. Any restrictions inside their input expressions still apply.

Two functions can be combined arithmetically. For $f$ and $g$, the sum, difference, product, and quotient are defined pointwise, and the domain of each is the intersection of the domains of $f$ and $g$ (the quotient also excludes any input where $g(x) = 0$).

A more powerful combination is composition, which feeds the output of one function into another.

On a domain symmetric about zero, symmetry of a graph corresponds to an algebraic test on $f(-x)$.

• Even: $f(-x) = f(x)$ — the graph is symmetric about the $y$-axis (e.g. $x^2 - 2$).
• Odd: $f(-x) = -f(x)$ — the graph is symmetric about the origin (e.g. $x^3 - 3x$).

A piecewise function applies different rules on different parts of its domain. The absolute-value function is the classic example — it is really two linear pieces joined at the point where the inside expression changes sign.

The greatest-integer (or floor) function $f(x) = \lfloor x \rfloor$ returns the greatest integer less than or equal to $x$, producing a staircase of horizontal steps — for example $\lfloor 2.7 \rfloor = 2$ and $\lfloor -1.2 \rfloor = -2$.