Defining a Function — Cheat sheet

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Key method

The notation $f:A\to B$ specifies the domain $A$ and codomain $B$ . The rule $f(x)$ tells how to assign outputs. The actual outputs form the range, which is a subset of $B$ .

Example

Let $f:\mathbb R\to\mathbb R$ have rule $f(x)=x^2$ . Then $f(-3)=9$ and $f(3)=9$ . These equal outputs do not violate the function definition: each input still has only one square. The equation $y^2=x$ describes a relation with two possible $y$ values for positive $x$ , unless a branch is selected.

Avoid this mistake

$f(x)$ is the output, not multiplication of $f$ by $x$ . The input symbol is a placeholder: $f(t)=t^2$ describes the same rule.