Functions

Quick Reference

Term	Definition / Condition	Symbol
Domain	Set of all valid inputs	$\text{Dom}(f)$
Codomain	Target set containing potential outputs	B in $f:A\to B$
Range	Set of all actual outputs $f(x)$	$\text{Rng}(f)$
Vertical Line Test	Intersects graph at most once	Test for function graphs

Definition of a Function

A function can be thought of as a machine or a computer program that assigns one output to every allowable input.

• If $y=f(x)$, we call x the independent variable, the argument of the function, or the input of the function, and y the dependent variable, the value of f at x, the image of x under f, or the output of the function.
• The definition of a function $f:A\to B$ does not restrict the nature of the elements of A and B, but in elementary calculus we assume that the elements of A and B are numbers; that is, A and B are subsets of the real numbers $ℝ$, unless stated otherwise.

Strictly speaking, $f(x)$ is the value of f at x, but we often talk about "the function $f(x)$" or "the function $y=f(x)$." For example, consider a function $f:ℝ\to ℝ$ that takes a number x and gives its square $x^2$. In this case, we can simply say: "the function $f(x)=x^2$."

Domain and Range

If a function is defined by an algebraic formula and the domain is not explicitly stated, the convention is that the domain is the natural domain: the set of all real numbers for which the formula is defined and yields a real number.

The Graph of a Function

The graph of a function $f$ is the set of all ordered pairs $(x,f(x))$ in the $xy$-plane such that $x\in \text{Dom}(f)$:

$$\text{Graph}(f)=\{(x,f(x))\mid x\in \text{Dom}(f)\}$$

Visualizing the graph allows us to observe domain (projected onto the x-axis) and range (projected onto the y-axis).

The Vertical Line Test