What Is a Function?

A function is a rule with one output for each allowed input

A function is a rule that takes an input, does something definite to it, and hands back exactly one output for every input it allows. The doubling rule is a function: feed it $5$ and it returns $10$, feed it $-3$ and it returns $-6$. The doubling rule never leaves any doubt about which number comes out. Everyday rules work the same way: “assign to each person their exact height right now” is a function, since every person has one height at this moment, but “assign to each height the person who is that tall” is not, since many different people can share a height.

That last requirement is the whole heart of the definition, so state it plainly: a function may never send one input to two different outputs. It is perfectly allowed for two different inputs to land on the same output. Squaring sends both $3$ and $-3$ to $9$, and that is fine, because each input still has a single, definite result. What is forbidden is the reverse, one input branching into two answers.

Here is a rule that breaks the requirement. Consider “send a number to a value whose square equals it.” Give it the input $9$ and there are two numbers whose square is $9$, namely $3$ and $-3$:

The input $9$ produces two outputs, so this rule is not a function. A function has to make up its mind, one input, one output.

Why function notation?

You already have a way to write outputs, the letter $y$. Function notation adds two things a bare $y$ leaves out: which rule produced the output, and which input was used.

Real problems rarely involve a single relationship. You might track a car’s distance, its speed, and its fuel all at once. If every output is just called $y$, then three different rules are wearing the same label, and an equation cannot say which $y$ it means. Function notation fixes this by naming the rule. Write $y = f(x)$ to say ”$f$ is the name of the rule, and $y$ is the output it produces.” For the rule $y = 2x + 1$, this becomes

read aloud as ”$f$ of $x$ equals $2x + 1$.” Different letters name different rules, so $f$, $g$, and $h$ can sit on the same page as three distinct machines.

Function notation also keeps the input attached to its output. Compute the output of $f$ at $x = 3$ and at $x = 5$:

Each statement carries its own input, so the output $7$ is never separated from the input $3$ that made it. That is the payoff: $f$ names the rule, so several rules can appear on the same page without confusion, and $f(3)$ pins an output to the exact input that produced it.

Reading and evaluating $f(x)$

Take the notation apart once so it never trips you up. In $f(x)$, the letter $f$ is the name of the function, and the quantity inside the parentheses is the input the rule acts on. The whole symbol $f(x)$ stands for the output. The name can be any letter that is convenient: $f$, $g$, and $h$ are common. And a descriptive letter such as $A$ for an area or $C$ for a cost is even better when it reminds you what the rule does.

One warning up front, because it is the most common misread of all. The parentheses in $f(x)$ do not mean multiplication. $f(3)$ is not ”$f$ times $3$”; it means “apply the rule $f$ to the input $3$.” There is no hidden product here, only the instruction to run the machine.

To evaluate a function at a particular input, substitute that number for $x$ everywhere it appears, then simplify. This is exactly the substitution you have done since your first algebra lesson, now wearing a cleaner label.

When a rule is not a function

The definition draws a sharp line: one input, one output. A rule sits on the wrong side of that line the instant a single input is tied to two different outputs. It is worth learning to spot this in the three ways relationships usually appear.

As a set of ordered pairs, a rule is a function when no input value is paired with two different output values. The set $\{(1, 2), (2, 4), (3, 6)\}$ is a function, and so is $\{(1, 5), (2, 5), (3, 5)\}$, where every input lands on the same output $5$ (repeated outputs are allowed). But $\{(1, 2), (1, 5)\}$ is not a function, because the input $1$ is paired with both $2$ and $5$.

As a table, the same test reads down the input column: if any input appears twice with different outputs beside it, the rule is not a function. As a graph, there is a quick visual version of the test, worth knowing because you have already plotted points and lines in the coordinate plane.

Every point on a graph is an input paired with its output: the input is the point’s horizontal position, the output its vertical position. A vertical line fixes one horizontal position, so it fixes one input, and every point where that line touches the graph is an output for that input. If the line touches the graph twice, that one input has two different outputs, exactly what a function forbids. If it touches once, the input has exactly one output, and if it misses the graph entirely, that input simply is not in the domain.

Domain and range

Two more words complete the vocabulary, and both are just names for collections of numbers.

The domain of a function is the set of all inputs it is allowed to take. The range is the set of all outputs it actually produces. Input goes with domain, output goes with range, and you can describe each in plain words without any special symbols. In this lesson, “number” means a real number, the kind you already plot on a number line.

Most rules built from adding, subtracting, and multiplying accept every number, so their domain is simply “all numbers.” Two situations shrink the domain, and both come from operations you already know cannot be done. Division by zero is undefined, so any input that would make a denominator zero is thrown out. And in this course the square root of a negative number is not a real number, so a square root forces its inside to stay zero or positive. For example, $g(x) = \dfrac{1}{x}$ has domain “all numbers except $0$,” because $x = 0$ would divide by zero. The square root rule $h(x) = \sqrt{x}$ has domain “all numbers greater than or equal to $0$,” because a negative input has no real square root.

The range takes a little more thought, because it depends on what the rule can actually reach. For $f(x) = x^2$, every output is a square, and a square is never negative, so no output can be below $0$. Every nonnegative number is reached too: to land exactly on some target $r \ge 0$, use the input $x = \sqrt{r}$, since $(\sqrt{r})^2 = r$. So the range is “all numbers greater than or equal to $0$,” the lower bound and every value above it, both accounted for.

Four ways to show the same function

A single function can be presented in four different outfits, and a fluent reader recognizes it in all of them. Take the rule “double the input and add one.” Here it is four ways at once.

As a rule (an equation), it is compact and ready to evaluate:

As a table, it lists sample inputs beside their outputs:

As a graph, it is the collection of points $(x, f(x))$ in the coordinate plane, which for this rule is the straight line you would get from $y = 2x + 1$. As words, it is the sentence “double the input, then add one.” All four describe the very same pairing of inputs with outputs. The rule and the words tell you how to produce any output; the table shows a handful of sample pairs, while the graph pictures every pair the function has, over whatever stretch of inputs is drawn.

Common mistakes