What a function is

A function is an unambiguous assignment: to every element of one set it assigns exactly one element of another set.

The easiest picture is a machine: you put a number in, a number comes out. The same number put in twice must give the same result — otherwise the machine is broken, and the assignment is not a function.

Argument, value, and the f(x) notation

The number going in is the argument, the number coming out is the value of the function at that argument:

We read $f(3) = 5$ as "the value of $f$ at the argument $3$ is $5$". This is not $f$ multiplied by $3$ — the bracket here says "compute the result for this input", not "multiply".

The question can also be turned around: we know the value and look for the argument. That is an equation:

Domain and range

The domain $D_f$ is the set of all arguments the function is defined for — what may go into the machine:

The range $R_f$ is the set of everything that comes out of it:

For $f(x) = 2x - 1$ the domain is every real number — any number can be doubled and reduced by one. It is not always so: the formula $f(x) = \frac{1}{x}$ makes no sense at $x = 0$, because we do not divide by zero, and $f(x) = \sqrt{x}$ makes no sense for negative numbers. The domain is therefore not a formality but the first thing to check when you look at a formula.

The domain can also be imposed by the wording of a problem: if $x$ counts the pupils in a class, then no formula, however pretty, will allow $x = -3$ or $x = 2.5$.

The graph of a function

The graph of a function is the set of points whose first coordinate is an argument and whose second one is the matching value:

It is the most convenient way to look at a function: everything about it is visible at once. A value is read in two moves — from the argument up to the graph, then left to the vertical axis:

The same path travelled the other way — from a value on the vertical axis right to the graph and down to the horizontal one — gives the preimage, that is the answer to "for which $x$ does the function take this value".

The vertical line test

Since one argument must have exactly one value, every vertical line can meet the graph of a function at most once:

This is why a circle is not the graph of a function: a vertical line through its interior hits two points, so one $x$ would have two different values of $y$. The reverse is allowed: two different arguments may share a value — the parabola above assigns the same value $3$ to both $-2$ and $2$ and remains a function.

Four ways to define a function

The same function can be described in several ways:

• by a formula — $f(x) = 2x - 1$;
• by a table — a finite list of argument–value pairs;
• by a graph — a set of points in the plane;
• in words — "assign to every number its double, less one".

A table is convenient but incomplete — it shows six pairs, while the function assigns a value to every real number. The formula and the graph carry all of it at once.

A zero, and the value at zero

Two numbers that are very easy to confuse:

• a zero of the function is an argument $x_0$ with $f(x_0) = 0$ — on the graph, a crossing with the $x$-axis;
• the value at zero is the number $f(0)$ — on the graph, the crossing with the $y$-axis.

For $f(x) = 2x - 1$ the zero is $x_0 = \frac{1}{2}$ and the value at zero is $f(0) = -1$. Two entirely different numbers with misleadingly similar names; we come back to both in the linear function.

Exercises

The questions cover both directions of the assignment: sometimes you are given the argument and asked for the value, sometimes given the value and asked which argument it came from. Type the number alone.

Common mistakes

• Reading $f(x)$ as a product — $f(3)$ is the value at three, not $f \cdot 3$.
• Confusing a zero with $f(0)$ — the first is an argument, the second a value.
• Losing a sign at a negative argument — in $f(-4)$ substitute $(-4)$, brackets included.
• Skipping the domain — $\frac{1}{x}$ fails at zero and $\sqrt{x}$ fails for negative numbers.
• Taking every curve for a graph of a function — a circle is not one; it fails the vertical line test.
• Mixing up domain and range — the domain lives on the horizontal axis, the range on the vertical one.