Domain of a Function

Article Contents

What is the Domain?
When the Domain is All Real Numbers
Restriction: Division by Zero
Restriction: Square Roots
Combining Restrictions
Interval Notation
Reading the Domain from a Graph
Common Mistakes
Interactive Exercises

1. What is the Domain? {#what-is-domain}

The domain of a function $f$ (written $D (f)$ ) is the set of all allowed inputs -- all values of $x$ for which $f (x)$ is defined and produces a real number.

D (f) = {x \in R ∣ f (x) is defined}

Think of it this way

The domain answers: "Which values of $x$ can I plug in?"

2. When the Domain is All Real Numbers {#domain-all-reals}

For many common functions, every real number is a valid input:

Function	Domain	Why
$f (x) = 2 x + 3$	$D (f) = R$	Linear -- no restrictions
$f (x) = x^{2} - 5 x + 1$	$D (f) = R$	Polynomial -- no restrictions
$f (x) = x^{3}$	$D (f) = R$	Polynomial -- no restrictions
$f (x) = 7$	$D (f) = R$	Constant -- no restrictions

Rule: Polynomials (expressions with

x

x^{2}

x^{3}

, ...) always have domain

R

3. Restriction: Division by Zero {#division-by-zero}

Division by zero is undefined. If the function has $x$ in a denominator, we must exclude values that make the denominator zero.

Example 1: $f (x) = \frac{1}{x}$

The denominator is $x$ . Set it equal to zero:

x = 0

So $x = 0$ is not allowed.

D (f) = R ∖ {0} = (- \infty, 0) \cup (0, + \infty)

Example 2: $f (x) = \frac{3}{x - 2}$

x - 2 = 0 ⟹ x = 2

D (f) = R ∖ {2}

Example 3: $f (x) = \frac{x + 1}{x ^{2} - 4}$

Factor the denominator:

x^{2} - 4 = (x - 2) (x + 2) = 0 ⟹ x = 2 or x = - 2

D (f) = R ∖ {- 2, 2}

4. Restriction: Square Roots {#square-roots}

The expression under a square root must be non-negative (greater than or equal to zero).

Example 1: $f (x) = x$

x \geq 0

D (f) = [0, + \infty)

Example 2: $f (x) = x - 3$

x - 3 \geq 0 ⟹ x \geq 3

D (f) = [3, + \infty)

Example 3: $f (x) = 5 - x$

5 - x \geq 0 ⟹ x \leq 5

D (f) = (- \infty, 5]

5. Combining Restrictions {#combining-restrictions}

Some functions have both a fraction and a square root.

Example: $f (x) = \frac{1}{x - 1}$

Two conditions:

Square root requires: $x - 1 \geq 0 ⟹ x \geq 1$

2. Denominator requires: $x - 1 \neq = 0 ⟹ x - 1 \neq = 0 ⟹ x \neq = 1$

Combining: $x > 1$ .

D (f) = (1, + \infty)

Example: $f (x) = \frac{x}{x - 4}$

Square root requires: $x \geq 0$
Denominator requires: $x \neq = 4$

D (f) = [0, 4) \cup (4, + \infty)

6. Interval Notation {#interval-notation}

We use intervals to describe the domain concisely:

Notation	Meaning
$(a, b)$	All $x$ with $a < x < b$ (endpoints excluded)
$[a, b]$	All $x$ with $a \leq x \leq b$ (endpoints included)
$[a, b)$	All $x$ with $a \leq x < b$
$(a, + \infty)$	All $x$ with $x > a$
$(- \infty, b]$	All $x$ with $x \leq b$
$R$	All real numbers = $(- \infty, + \infty)$
$R ∖ {c}$	All real numbers except $c$

Union of intervals

When the domain has a "gap", we use $\cup$ (union):

D (f) = (- \infty, 2) \cup (2, + \infty)

This means all real numbers except $x = 2$ .

7. Reading the Domain from a Graph {#domain-from-graph}

On a graph, the domain is the set of all $x$ -values where the graph exists.

Look at the horizontal extent of the graph -- from the leftmost to the rightmost point.

8. Common Mistakes {#common-mistakes}

Mistake	Correction
Forgetting to check the denominator	Always set denominator $\neq = 0$
Writing $x - 3 \geq 0$ instead of $x - 3 \geq 0$	The condition goes on the expression under the root
Using $>$ instead of $\geq$ for square roots	$0 = 0$ is defined, so include the boundary
Ignoring the case $\frac{1}{\dots}$	Here the expression under the root must be strictly positive ( $> 0$ )

9. Interactive Exercises {#interactive-exercises}

Practice finding domains:

Summary

Function type	Domain rule
Polynomial ( $x^{2} + 3 x - 1$ )	$D (f) = R$
Fraction ( $\frac{a}{g ( x )}$ )	$g (x) \neq = 0$
Square root ( $g (x)$ )	$g (x) \geq 0$
Root in denominator ( $\frac{a}{g ( x )}$ )	$g (x) > 0$

Introduction to Functions -- what is a function
Range of a Function -- which outputs are possible
Functions and Graphs -- visualising functions
Inverse Proportion -- a function with $D (f) = R ∖ {0}$

Domain of a Function

Domain of a Function

Article Contents

1. What is the Domain? {#what-is-domain}

Think of it this way

2. When the Domain is All Real Numbers {#domain-all-reals}

3. Restriction: Division by Zero {#division-by-zero}

Example 1: f(x)=x1​

Example 2: f(x)=x−23​

Example 3: f(x)=x2−4x+1​

4. Restriction: Square Roots {#square-roots}

Example 1: f(x)=x​

Example 2: f(x)=x−3​

Example 3: f(x)=5−x​

5. Combining Restrictions {#combining-restrictions}

Example: f(x)=x−1​1​

Example: f(x)=x−4x​​

6. Interval Notation {#interval-notation}

Union of intervals

7. Reading the Domain from a Graph {#domain-from-graph}

8. Common Mistakes {#common-mistakes}

9. Interactive Exercises {#interactive-exercises}

Summary

Related Articles

Example 1: $f (x) = \frac{1}{x}$

Example 2: $f (x) = \frac{3}{x - 2}$

Example 3: $f (x) = \frac{x + 1}{x ^{2} - 4}$

Example 1: $f (x) = x$

Example 2: $f (x) = x - 3$

Example 3: $f (x) = 5 - x$

Example: $f (x) = \frac{1}{x - 1}$

Example: $f (x) = \frac{x}{x - 4}$