Functions & Function Notation

Evaluating f(x), the single-output rule, domain, and range.

The idea

A function is a rule that assigns to each input exactly one output. The requirement is exactly one: a rule that offered some input two different outputs would not determine an output at all. Several inputs may share one output; only the reverse is forbidden.

We name the rule with a letter such as $f$, and write $f(x)$, read aloud as $f$ of $x$, for the output $f$ assigns to the input $x$. The parentheses enclose an input — they do not indicate multiplication. Thus $f(x) = 3x + 2$ defines the rule that triples its input and adds $2$, and to evaluate $f$ at a number we replace every $x$ by that number and simplify.

The domain of a function is the set of inputs it accepts, and its range is the set of outputs it produces. Any input that breaks the rule — by forcing a division by zero, for example — is excluded from the domain.

Ways to work on it

• Walkthrough. Evaluate f(x), test the single-output rule, and read off a range.
• Practice. Evaluate a function at a given input.
• Hardest. Find a domain restriction, then evaluate inside the domain.

Not sure where to start? Take the ten-question placement test.