SL 2.2—Functions, domains, range, graphs

A function is a mathematical rule that takes an input value and produces exactly one output value. Think of it as a machine: you feed in a number $x$, and the machine applies a rule to produce an output $f(x)$.

Function: A function $f$ is a rule that assigns to each input value $x$ in its domain exactly one output value, written $f(x)$.

The critical requirement is the one output per input rule. If a single input could produce two different outputs , for example, $f(x)=\pm \sqrt{x}$ , then the relationship is merely a relation, not a function.

The standard way to write a function is $f(x)$, where:

• $f$ is the name of the function
• $x$ is the input variable (also called the argument)
• The expression defining the rule is written on the right-hand side of an equation

Other letters are commonly used depending on context. For instance:

• $v(t)$ , velocity as a function of time
• $C(n)$ , cost as a function of number of items
• $h(t)$ , height as a function of time

An alternative notation you may encounter is the mapping notation:
$f:x\to y$
This means exactly the same as $f(x)=y$ , the function $f$ maps input $x$ to output $y$.

Evaluating a function means substituting a specific number in place of $x$: