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The Domain of a Function

The Basic Concept of a Domain

The domain (Domain) is the set of all values of the independent variable for which the function is meaningful, usually written $D(f)$ or $D$ .

For a function $y = f(x)$ , the domain is the set of all $x$ values for which the function expression makes sense.

Methods for Finding the Domain of a Function

1. The Domain of a Rational Function

For a rational function, the denominator must not be zero.

Example 1: find the domain of $f(x) = \frac{1}{x-2}$ .

Solution: for the function to be meaningful we need $x - 2 \neq 0$ , that is, $x \neq 2$ .

Hence the domain is $(-\infty, 2) \cup (2, +\infty)$ .

2. The Domain of a Radical Function

For an even root, the radicand must be non-negative.

Example 2: find the domain of $f(x) = \sqrt{x-1}$ .

Solution: for the function to be meaningful we need $x - 1 \geq 0$ , that is, $x \geq 1$ .

Hence the domain is $[1, +\infty)$ .

Example 3: find the domain of $f(x) = \sqrt{4-x^2}$ .

Solution: for the function to be meaningful we need $4 - x^2 \geq 0$ , that is, $x^2 \leq 4$ .

Solving the inequality: $-2 \leq x \leq 2$ .

Hence the domain is $[-2, 2]$ .

3. The Domain of a Logarithmic Function

For a logarithmic function, the argument must be positive.

Example 4: find the domain of $f(x) = \log_2(x+3)$ .

Solution: for the function to be meaningful we need $x + 3 > 0$ , that is, $x > -3$ .

Hence the domain is $(-3, +\infty)$ .

4. The Domain under Combined Conditions

When a function contains several restrictions, all of them have to be satisfied at once.

Example 5: find the domain of $f(x) = \frac{\sqrt{x-1}}{x-3}$ .

Solution: for the function to be meaningful we need to satisfy at the same time:

$x - 1 \geq 0$ (the condition for the radical)
$x - 3 \neq 0$ (the condition for the fraction)

that is:

$x \geq 1$
$x \neq 3$

Hence the domain is $[1, 3) \cup (3, +\infty)$ .

Ways of Writing a Domain

A domain can be written in any of the following ways:

Interval notation: $[a, b]$ , $(a, b)$ , $[a, +\infty)$ and so on
Set-builder notation: $\{x | x \geq 1, x \neq 3\}$
Inequality notation: $x \geq 1$ and $x \neq 3$

The Domains of Common Functions

Type of function	General form	Condition on the domain
Polynomial function	$f(x) = a_nx^n + \cdots + a_1x + a_0$	$\mathbb{R}$ (all real numbers)
Rational function	$f(x) = \frac{P(x)}{Q(x)}$	$Q(x) \neq 0$
Even root function	$f(x) = \sqrt[2n]{g(x)}$	$g(x) \geq 0$
Odd root function	$f(x) = \sqrt[2n+1]{g(x)}$	$\mathbb{R}$
Logarithmic function	$f(x) = \log_a g(x)$	$g(x) > 0$ , $a > 0$ and $a \neq 1$

Exercises

Exercise 1

Find the domain of $f(x) = \frac{1}{x-2} + \sqrt{x+1}$ .

Reference Answer

the domain of a functiona composite function

Approach:

The two parts have to be considered separately:

the fraction $\frac{1}{x-2}$ : the denominator must not be zero
the radical $\sqrt{x+1}$ : the radicand must be non-negative

Steps:

For $\frac{1}{x-2}$ , we need $x-2 \neq 0$ , that is, $x \neq 2$
For $\sqrt{x+1}$ , we need $x+1 \geq 0$ , that is, $x \geq -1$
Combine the two conditions: $x \geq -1$ and $x \neq 2$

Answer: The domain is $[-1, 2) \cup (2, +\infty)$

Exercise 2

Find the domains of the following functions:

$f(x) = \log_2(3x - 6)$
$g(x) = \sqrt[3]{x - 1}$
$h(x) = \frac{\sqrt{x + 2}}{x^2 - 9}$

Reference Answer

the domain of a functionlogarithmic functionsodd roots

Approach:

Solve each one according to the domain requirements of a logarithmic function, of an odd root function, and of a composite function respectively.

Steps:

For $f(x) = \log_2(3x - 6)$ :
- a logarithmic function requires the argument to be positive
- $3x - 6 > 0$
- $3x > 6$
- $x > 2$
- the domain is $(2, +\infty)$
For $g(x) = \sqrt[3]{x - 1}$ :
- an odd root function puts no restriction on the radicand
- the domain is $\mathbb{R}$
For $h(x) = \frac{\sqrt{x + 2}}{x^2 - 9}$ :
- the radical requires $x + 2 \geq 0$ , that is, $x \geq -2$
- the denominator requires $x^2 - 9 \neq 0$ , that is, $x \neq \pm 3$
- combining: $x \geq -2$ and $x \neq 3$ (the value $x = -3$ is already excluded by $x \geq -2$ )
- the domain is $[-2, 3) \cup (3, +\infty)$

Answer:

$(2, +\infty)$
$\mathbb{R}$
$[-2, 3) \cup (3, +\infty)$

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$D(f)$	Mathematical symbol	D of f	The domain of the function $f$
$D$	Mathematical symbol	D	The domain of the function
$y = f(x)$	Mathematical formula	y equals f of x	The general way of writing a function
$\sqrt{x}$	Mathematical symbol	square root of x	The square root of $x$
$\sqrt[n]{x}$	Mathematical symbol	n-th root of x	The $n$ -th root of $x$
$\log_a x$	Mathematical symbol	logarithm base a of x	The logarithm of $x$ to base $a$
$\mathbb{R}$	Mathematical symbol	double-struck R (Real numbers)	Denotes the set of real numbers
$[a, b]$	Mathematical symbol	closed interval	Interval notation including the endpoints
$(a, b)$	Mathematical symbol	open interval	Interval notation excluding the endpoints
$[a, b)$	Mathematical symbol	half-open interval	Interval notation with the left end closed and the right end open
$(a, b]$	Mathematical symbol	half-open interval	Interval notation with the left end open and the right end closed
$(-\infty, +\infty)$	Mathematical symbol	infinite interval	Interval notation for all real numbers
$\cup$	Mathematical symbol	union symbol	Denotes the union of two sets

Ways of Representing a Function

The Range of a Function

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