Introduction, Domain & Range

Video lesson · from 4:42

One lesson video covers all of Functions & Graphs, so it opens at 4:42 for this topic — not from the beginning.

Theory

Introduction To Functions & Sets

A function is a rule which connects one set of numbers to another.

A function can be expressed in the following forms:

Table
Graph
Arrow diagram
Formula

Notation

$f(x) = \dots$ i.e. f of x

$g(u) = \dots$ i.e. g of u

A function is a relation between two sets where each member of the first set is related to only one member of the second set.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Division by Zero

Since we cannot divide any number by zero, the domains of functions involving fractions must exclude numbers which would result in a denominator of zero.

$f(x) = \frac{3}{x} \implies x \neq 0$

$f(x) = \frac{2}{10-x} \implies x \neq 10$

$f(x) = \frac{2x+1}{x^2-4x} \implies x \neq 0, x \neq 4$

Even Roots

We cannot evaluate an even root of a negative number. The domain must exclude numbers which would result in negative numbers under the root.

$f(x) = \sqrt{x} \implies x \geq 0$

$f(x) = \sqrt{x-4} \implies x \geq 4$

$f(x) = \frac{2x+1}{\sqrt{x-4}} \implies x > 4$

Range of a Function

The range of a function is the spread of possible y-values (outputs).

Function	Range
$f(x) = \cos x$	$-1 \leq f(x) \leq 1$
$f(x) = x^2 + 4$	$f(x) \geq 4$
$f(x) = \sqrt{x}$	$f(x) \geq 0$
$f(x) = \sqrt{x} - 2$	$f(x) \geq -2$
$f(x) = x^3$	$f(x) \in \mathbb{R}$

⚠️ Common Examiner Traps

State the condition, not just the domain. Marks are routinely lost by giving the domain without saying why it is restricted. Name the value that breaks it: a denominator of zero, or a negative under a square root.
Denominators: for $f(x) = \frac{1}{x-3}$ the function is undefined at $x = 3$ , so the domain is all real $x$ except 3. Set the denominator equal to zero to find it.
Square roots: for $\sqrt{g(x)}$ you need $g(x) \geq 0$ — solve that inequality rather than guessing.
Domain is inputs, range is outputs: writing one when the question asks for the other loses the mark even when the numbers are right.