Functions: definition and notation

A function is the most important kind of relation in all of mathematics. It captures the idea of a deterministic rule: feed in one input, get exactly one output. From here every later topic , limits, derivatives, integrals , is built.

Definitions

Let $A$ and $B$ be non-empty sets. A function $f$ from $A$ to $B$ is a relation $f \subseteq A \times B$ such that

1. (Total) every $a \in A$ appears as the first coordinate of some pair, and
2. (Single-valued) if $(a, b) \in f$ and $(a, c) \in f$, then $b = c$.

We write $f : A \to B$ and, for each $a \in A$, denote the unique $b$ paired with $a$ by $f(a)$. So $b = f(a)$ is just a short form of "$(a, b) \in f$".

• $A$ is the domain of $f$.
• $B$ is the codomain of $f$.
• $\{f(a) : a \in A\}$ is the range of $f$ , a subset of $B$.

The two conditions say: "every input has an output" and "no input has two outputs".

Three ways to specify a function

1. By a formula. $f(x) = x^2$, $f : \mathbb{R} \to \mathbb{R}$. The rule pairs each $x$ with $x^2$.

2. By a table or listing. For finite $A$ we can list $f(a)$ for each $a$.

3. By a graph. For real-valued functions, the graph $\{(x, f(x)) : x \in A\}$ in $\mathbb{R}^2$.

The vertical line test

A subset $S \subseteq \mathbb{R}^2$ is the graph of a function $y = f(x)$ if and only if every vertical line $x = c$ meets $S$ in at most one point (and exactly one point if $c$ is in the domain). This is just the single-valued rule, picture-side.

Examples:

• $y = x^2$: every vertical line cuts the parabola once. Function.
• $x^2 + y^2 = 1$: most vertical lines cut the circle in two points. Not a function.
• $y = \sqrt{x}$ (taking the non-negative root): every vertical line in $[0, \infty)$ meets it once. Function.

When two functions are equal

Two functions $f$ and $g$ are equal if and only if:

1. They have the same domain.
2. $f(a) = g(a)$ for every $a$ in the domain.

So $f(x) = x$ on $\mathbb{R}$ and $g(x) = x$ on $\mathbb{Z}$ are different functions, even though their formulas look the same.

Examples of "function or not"

• $f : \mathbb{R} \to \mathbb{R}$, $f(x) = 2x + 3$: function.
• $f : \mathbb{R} \to \mathbb{R}$, $f(x) = \frac{1}{x}$: not a function from $\mathbb{R}$ to $\mathbb{R}$ , undefined at $x = 0$. But it is a function from $\mathbb{R} \setminus \{0\}$ to $\mathbb{R}$.
• $f(x) = \sqrt{x}$: function from $[0, \infty)$ to $\mathbb{R}$.
• The "rule" $f(x) = \pm \sqrt{x}$ pairs $x = 4$ with both $2$ and $-2$ , not a function.

Worked examples

Example 1. Decide whether $\{(1, 2), (2, 3), (3, 4)\}$ is a function from $\{1, 2, 3\}$ to $\{2, 3, 4, 5\}$. State its domain, codomain, range.

Every element of $\{1, 2, 3\}$ appears exactly once as a first coordinate, so this is a function. Domain $= \{1, 2, 3\}$, codomain $= \{2, 3, 4, 5\}$, range $= \{2, 3, 4\}$.

Example 2. Is $\{(1, 2), (2, 3), (1, 4)\}$ a function on $\{1, 2\}$?

No , the input $1$ is paired with two outputs $2$ and $4$. Single-valuedness fails.

Example 3. Is the curve $y^2 = x$ in the plane the graph of a function $y = f(x)$?

No. For $x = 4$, both $y = 2$ and $y = -2$ satisfy $y^2 = x$. The vertical line $x = 4$ meets the curve at two points.

Example 4. Find $f(0), f(1), f(-2)$ for $f(x) = x^2 - 3x + 4$.

$f(0) = 0 - 0 + 4 = 4$. $f(1) = 1 - 3 + 4 = 2$. $f(-2) = 4 + 6 + 4 = 14$.

Example 5 (harder). A function $f : \mathbb{R} \to \mathbb{R}$ satisfies $f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$, and $f(1) = 3$. Find $f(0), f(2), f(-1), f(5)$.

Putting $x = y = 0$: $f(0) = 2 f(0)$, so $f(0) = 0$.

Putting $y = -x$: $f(0) = f(x) + f(-x) = 0$, so $f(-x) = -f(x)$.

Now $f(2) = f(1) + f(1) = 6$, $f(-1) = -3$, $f(5) = f(2) + f(2) + f(1) = 6 + 6 + 3 = 15$.

(In fact this forces $f(n) = 3n$ for every integer $n$, and with mild extra assumptions, $f(x) = 3x$ for every real $x$.)

Try it yourself

1. Decide whether each is a function: (a) $\{(1, 2), (2, 4), (3, 6)\}$ from $\{1, 2, 3\}$ to $\mathbb{N}$; (b) $\{(1, 2), (1, 3)\}$ from $\{1, 2\}$ to $\mathbb{N}$; (c) $\{(1, 1), (2, 1), (3, 1)\}$ from $\{1, 2, 3\}$ to $\mathbb{N}$.
2. Find $f(2), f(-3), f(0)$ for $f(x) = x^3 - 2x + 1$.
3. Does $y = |x|$ define a function from $\mathbb{R}$ to $\mathbb{R}$?
4. Does the circle $x^2 + y^2 = 25$ define $y$ as a function of $x$?
5. Are $f(x) = x$ on $\mathbb{R}$ and $g(x) = \frac{x^2}{x}$ on $\mathbb{R} \setminus \{0\}$ equal functions? Explain.
6. State the codomain and find the range of $f : \mathbb{R} \to \mathbb{R}$, $f(x) = x^2$.
7. If $f(x) = ax + b$ with $f(1) = 5, f(2) = 7$, find $a$ and $b$.
8. Plot the graph of $f(x) = 2x - 1$ for $x \in [-1, 2]$. State its range.
9. Show that $f(x) = \frac{x}{1 + x^2}$ is a function from $\mathbb{R}$ to $\mathbb{R}$ and find its value at $x = 0, 1, -1$.
10. If $f(x + 1) = x^2 - 3x + 2$, find $f(x)$.
11. If $f(x) = x^2$, evaluate $\frac{f(x+h) - f(x)}{h}$ for $h \ne 0$ and simplify.
12. Sketch a curve that satisfies the vertical line test on $[-2, 2]$ but is not a polynomial.

Pitfalls / Tricks

• A function must be defined for every input in the domain. Always state the domain explicitly: $f(x) = 1/x$ "on $\mathbb{R}$" is incorrect; "on $\mathbb{R} \setminus \{0\}$" is correct.
• Two functions can have the same formula yet be unequal if their domains differ.
• The range is a subset of the codomain , never confuse them.
• Insight. A function is a machine: one input, one output, no exceptions. If a rule sometimes returns multiple values, sometimes none, it is a relation, not a function. Beginning with this distinction prevents endless confusion later.