This is a beta course, so its structure, chapters, and examples may continue to change.

The Definition of a Function

Definition

Let there be two variables $x$ and $y$ . If for every value of $x$ in some range $D$ , the variable $y$ has a unique determined value corresponding to it, then $y$ is called a function of $x$ , written

y = f(x)

where $x$ is called the independent variable, $y$ is called the dependent variable, and $D$ is called the domain of the function.

What does the word "function" suggest about a function, and what everyday objects behave like functions?

The English word function comes from Latin functio, a performance or an execution — the word carries the sense of something being carried out, and a function does exactly that: it takes an input and performs a definite operation on it.

The mathematical word also keeps a sense of “containing”: a function is like a box, and for every input value (the independent variable) one can “take out” a unique determined output value (the dependent variable).

The same picture shows up all over ordinary life. A vending machine takes a coin and a button press and hands back a drink; an oven takes a dish and a setting and hands back a baked meal; a vending machine at a different price hands back a different drink. Each of these is a function in the mathematical sense: a rule that pairs each input with exactly one output.

Exercises

Exercise 1

Judge whether each of the following relations is a function:

$y = \pm \sqrt{x}$
$y = x^2$
$x^2 + y^2 = 1$

Reference Answer

function definition

Approach:

To judge whether a relation is a function, the core is to test single-valuedness: given $x$ , does $y$ have a unique determined value? Check each of the three relations against that standard.

Steps:

$y = \pm \sqrt{x}$ is not a function, because for $x > 0$ the value of $y$ has two possibilities (the positive and negative roots), which violates single-valuedness.
$y = x^2$ is a function, because for every real $x$ there is a unique $y$ corresponding to it.
$x^2 + y^2 = 1$ is not a function, because for $-1 < x < 1$ the value of $y$ has two possibilities (the positive and negative roots), which violates single-valuedness.

Answer:

Only $y = x^2$ is a function.

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$x$	Mathematical symbol	x	The independent variable, the input of the function
$y$	Mathematical symbol	y	The dependent variable, the output of the function
$f(x)$	Mathematical symbol	f of x	The notation for a function with independent variable $x$
$D$	Mathematical symbol	D	The domain of the function

The Basic Concepts of Functions

Ways of Representing a Function

Found an issue with this content?

Tell us what could be improved.

Unlock Full Course ·

Your membership covers this course — no need to buy it again.

Login Required to Purchase

Payment Failed

Payment Successful!

Price History