Functions, domain and range

What a function is, the meaning of domain and range, how to find a range (and why a quadratic's is bounded), and one-to-one vs many-to-one mappings.

Pure · Algebra & functions

Functions, domain and range

What a function is, the meaning of domain and range, how to find a range (and why a quadratic's is bounded), and one-to-one vs many-to-one mappings.

Why it works

A function is a rule that takes each input to exactly one output. We write

f(x)

for the output at

x

, or

f: x \mapsto 2x + 1

. Two sets go with it:

the domain — the inputs the function is allowed to take ( $x$ -values),
the range — the outputs it actually produces ( $y$ -values).

Finding a range is reading off how high and low the outputs go *over the given domain*. A linear function on all of

\mathbb{R}

produces every value, so its range is all of

\mathbb{R}

. A quadratic is different: it turns around at its vertex, so its outputs are bounded on one side. Complete the square to find the vertex — e.g.

f(x) = x^2 - 6x + 11 = (x - 3)^2 + 2

has a minimum of

2

, so its range is

f(x) \ge 2

.

Watch the domain. If the domain is restricted, the range changes. For

f(x) = x^2 - 4x + 1

0 \le x \le 5

, the vertex

(2, -3)

gives the minimum, but the maximum is at an endpoint — check

f(0) = 1

and

f(5) = 6

— so the range is

-3 \le f(x) \le 6

. Always test the vertex and the endpoints.

Mappings: one-to-one or many-to-one. A function is one-to-one if different inputs always give different outputs (like

f(x) = x^3

). It is many-to-one if two inputs can share an output — e.g.

f(x) = x^2

has

f(2) = f(-2) = 4

. (This matters later: only one-to-one functions have an inverse.) A rule that gives two outputs for one input — like

\pm\sqrt{x}

— isn't a function at all.