What Is a Function?

A function is a rule with one output for each input

A function is a rule that takes an input, does something definite to it, and hands back exactly one output. 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, because every person has one height at this moment. Turn that around, though, and consider “assign to each height the person who is that tall.” That rule is not a function, because plenty of different people share a height, so a single input would have many outputs.

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 a bare $y$ is not enough

You already have a way to write outputs, the letter $y$, so why bother with anything new? Because $y$ quietly drops two pieces of information you often need.

The first is the name of the rule. 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 you cannot tell in an equation which $y$ you mean. Function notation gives each rule its own name. Call one rule $f$, another $g$, another $h$, and now $f$, $g$, and $h$ are three distinct machines you can discuss without confusion.

The second is the input that was used. Start from $y = 2x + 1$ and compute the output when $x = 3$:

You get $y = 7$, but the symbol "$y = 7$" has already forgotten that the input was $3$. Compute the output at $x = 5$ as well and you get a second "$y = 11$", and nothing in the notation ties either output back to the input that made it. Once a page has several of these, the bookkeeping falls apart.

Function notation fixes both problems in one stroke. Write the same rule as

read aloud as ”$f$ of $x$ equals $2x + 1$.” The letter $f$ names the rule, and whatever you put in the parentheses is the input. Now the output at $3$ is written $f(3)$, and you can record the whole fact on one line, input and output together:

Each statement carries its own input, so no output is ever orphaned. That is the payoff: $f(x)$ names the rule so several rules can coexist, and $f(3)$ pins the 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.

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.

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 the range is “all numbers greater than or equal to $0$.” The input $x$ ranges over every number, but the outputs only ever fill the nonnegative side.

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, while the table and the graph show you particular ones.

Common mistakes