Domain and range

Overview

The domain of a relation or function is the set of allowed inputs. The range is the set of outputs produced from those inputs.

This topic matters because a rule is not complete until we know where it can be used. For example, $f(x)=\frac{1}{x-2}$ is not defined at $x=2$, so its domain must exclude $2$. Domain and range also support inverse functions and graph interpretation.

The safest way to think is: domain comes before the rule is used, because it tells us what inputs are allowed. Range comes after the rule is used, because it tells us what outputs actually appear. This before-and-after idea is the bridge between tables, formulas, and graphs.

Prerequisites

• Sets, subsets, operations with sets, and Venn diagrams of two sets - Domain and range are sets.
• Relations and functions - Domain and range describe the input and output sides of relations and functions.
• Algebraic expressions and equations - Restrictions often come from algebraic rules.
• Inequalities in one unknown - Domains may be written using inequality notation.
• Graphs of relations and functions - Graphs show possible inputs and outputs visually.

Learning Scope

This chapter covers domain and range from ordered pairs, tables, simple rules, restricted rational functions, and piecewise functions. It also introduces how to describe domain and range using set notation, inequality notation, and short verbal statements, with checking routines for restrictions and repeated values.

This page does not fully teach inverse relations or graph drawing. It uses those ideas only when they help explain how domain and range behave.

Subtopics

Domain From Ordered Pairs

For a relation written as ordered pairs, the domain is the set of first components.

If:

$$ R=\{(1,4),(2,6),(3,8),(4,10)\} $$

then:

$$ \text{Domain}=\{1,2,3,4\} $$

Why this works: in each pair $(x,y)$, the first component is the value that is allowed to enter the relation. So the domain is found by reading the first position only.

Checking routine:

1. Circle or list the first value in every ordered pair.
2. Remove repeated values.
3. Write the remaining values as a set.

Range From Ordered Pairs

The range is the set of second components.

For:

$$ R=\{(1,4),(2,6),(3,8),(4,10)\} $$

the range is:

$$ \text{Range}=\{4,6,8,10\} $$

If an output repeats, list it once in the range.

Why this works: a set records membership, not the number of times a value appears. If the output $6$ appears twice, it is still one element of the range.

Checking routine:

1. Circle or list the second value in every ordered pair.
2. Remove repeated values.
3. Write the remaining values as a set.

Domain And Range From A Table

In a table, the input column gives the domain and the output column gives the range.

| $x$ | $f(x)$ | |---:|---:| | $-2$ | $4$ | | $-1$ | $1$ | | $0$ | $0$ | | $1$ | $1$ | | $2$ | $4$ |

The domain is:

$$ \{-2,-1,0,1,2\} $$

The range is:

$$ \{0,1,4\} $$

Notice that the table has five input values but only three range values because some outputs repeat. The table gives the actual values being used, so the domain is not "all numbers"; it is only the listed inputs unless the question states a wider domain.

Restricted Values

Some rules exclude values that make an expression undefined. The most common Form III restriction is division by zero.

For:

$$ f(x)=\frac{1}{x-2} $$

the denominator must not be zero:

$$ x-2 \ne 0 $$

so:

$$ x \ne 2 $$

Therefore the domain is all real numbers except $2$.

Why this works: division by zero has no value in ordinary arithmetic. The expression $x-2$ is the denominator, so any input that makes $x-2=0$ must be removed before the function can be used.

For rational expressions, use this domain routine:

1. Find the denominator.
2. Set the denominator not equal to zero.
3. Solve the restriction.
4. State the domain by excluding the restricted value or values.

Range Of A Rational Function

For:

$$ f(x)=\frac{1}{x-2} $$

the output cannot be $0$ because a fraction with numerator $1$ cannot equal $0$.

So the range is all real numbers except $0$:

$$ y \ne 0 $$

This range idea is different from the domain restriction. The domain restriction came from the denominator. The range restriction comes from asking which $y$-values can be produced. A fraction with numerator $1$ cannot produce $0$, so $0$ is excluded from the outputs.

Domain And Range Of Piecewise Functions

For:

$$ f(x)= \begin{cases} -2, & 0 < x \le 5 \\ x+1, & -6 \le x < 0 \end{cases} $$

the domain is the union of the input intervals:

$$ -6 \le x < 0 \quad \text{or} \quad 0 < x \le 5 $$

The range comes from outputs:

• For $-6 \le x < 0$, $x+1$ gives outputs from $-5$ up to but not including $1$.
• For $0 < x \le 5$, the output is always $-2$, which is already inside that interval of outputs.

So the range is:

$$ -5 \le y < 1 $$

Key insight: Domain comes from allowed inputs; range comes from produced outputs. Do not mix the two.

Endpoint bridge: if an input endpoint is included, its output may affect the range. If an input endpoint is excluded, its output may be approached but not included. This is why endpoint signs must be carried carefully from the domain interval to the output interval.

Key Terms

• Domain: The set of all allowed input values.
• Range: The set of all output values that are actually produced.
• Input: A value used in a relation or function.
• Output: A value obtained from a relation or function.
• Restriction: A value or interval that is not allowed.
• Undefined: A value is undefined when the rule cannot produce a valid result, such as division by zero.
• Interval: A continuous set of values described by inequalities.

Worked Examples

Example 1: Domain And Range From Ordered Pairs

Find the domain and range of:

$$ R=\{(-1,2),(0,2),(1,4),(2,6)\} $$

The domain is the set of first components:

$$ \{-1,0,1,2\} $$

The range is the set of second components, with repeated values listed once:

$$ \{2,4,6\} $$

Check:

• Domain uses the first components: $-1,0,1,2$.
• Range uses the second components: $2,2,4,6$.
• The repeated output $2$ is written once.

Example 2: Domain And Range Of A Rational Function

Find the domain and range of:

$$ f(x)=\frac{1}{x-2} $$

For the domain, avoid zero in the denominator:

$$ \begin{aligned} x-2 &\ne 0 \\ x &\ne 2 \end{aligned} $$

For the range, let $y=\frac{1}{x-2}$. This expression can never equal $0$, so:

$$ y \ne 0 $$

Therefore:

$$ \text{Domain: } x \ne 2,\quad \text{Range: } y \ne 0 $$

Check:

• Try the excluded input: $x=2$ gives $\frac{1}{0}$, so it must not be in the domain.
• Try the excluded output: $\frac{1}{x-2}=0$ has no solution because the numerator is $1$, so $0$ must not be in the range.

Example 3: Domain And Range Of An Inverse Context

Suppose $f(x)=3x^2$ is used for whole-number inputs $x=0,1,2,3,4$.

The domain is:

$$ \{0,1,2,3,4\} $$

The outputs are:

$$ 0,\ 3,\ 12,\ 27,\ 48 $$

So the range is:

$$ \{0,3,12,27,48\} $$

If an inverse relation is formed, the old range becomes the new domain, and the old domain becomes the new range.

Check:

The original relation contains $(0,0),(1,3),(2,12),(3,27),(4,48)$. Its inverse would contain $(0,0),(3,1),(12,2),(27,3),(48,4)$, so the old outputs are now first components.

Example 4: Domain From A Denominator With Two Restrictions

Find the domain of:

$$ h(x)=\frac{x+1}{x^2-9} $$

The denominator must not be zero:

$$ x^2-9 \ne 0 $$

Factor the denominator:

$$ (x-3)(x+3)\ne 0 $$

So neither factor may be zero:

$$ x-3\ne 0 \quad \text{and} \quad x+3\ne 0 $$

Therefore:

$$ x\ne 3 \quad \text{and} \quad x\ne -3 $$

The domain is all real numbers except $-3$ and $3$.

Check:

Substituting $x=3$ gives denominator $9-9=0$. Substituting $x=-3$ also gives denominator $9-9=0$. Both must be excluded.

Example 5: Domain And Range Of A Piecewise Function

Let:

$$ p(x)= \begin{cases} x+2, & -3 \le x < 1 \\ 5, & 1 \le x \le 4 \end{cases} $$

First find the domain by joining the input intervals:

$$ -3 \le x < 1 \quad \text{or} \quad 1 \le x \le 4 $$

Together these cover:

$$ -3 \le x \le 4 $$

Now find the range.

For the first branch, $p(x)=x+2$:

• When $x=-3$, $p(x)=-1$, and this endpoint is included.
• As $x$ gets close to $1$ from the left, $p(x)$ gets close to $3$, but $x=1$ is not included in this branch.

So the first branch gives:

$$ -1 \le y < 3 $$

The second branch gives the constant output:

$$ y=5 $$

Therefore the range is:

$$ -1 \le y < 3 \quad \text{or} \quad y=5 $$

Check:

Do not fill the gap between $3$ and $5$. No input in the given rule produces outputs such as $4$ or $4.5$.

Example 6: Finite Domain For A Quadratic Rule

For $f(x)=x^2-1$ with domain $\{-2,-1,0,1,2\}$, find the range.

Make a table:

| $x$ | Working | $f(x)$ | | ---: | --- | ---: | | $-2$ | $(-2)^2-1=4-1$ | $3$ | | $-1$ | $(-1)^2-1=1-1$ | $0$ | | $0$ | $0^2-1=0-1$ | $-1$ | | $1$ | $1^2-1=1-1$ | $0$ | | $2$ | $2^2-1=4-1$ | $3$ |

The outputs are $3,0,-1,0,3$, so the range is:

$$ \{-1,0,3\} $$

Check:

The range has fewer values than the domain because opposite inputs such as $-2$ and $2$ give the same output.

Common Mistakes

• Listing outputs as the domain. Correction: domain is the input set. Warning sign: you are reading the second column or second component first.
• Listing inputs as the range. Correction: range is the output set. Warning sign: your answer for range matches the $x$-values exactly without checking the rule.
• Repeating values unnecessarily in a set. Correction: each set element is written once. Warning sign: your range contains values such as $\{2,2,4\}$.
• Forgetting to exclude a value that makes a denominator zero. Correction: solve denominator $\ne 0$. Warning sign: the function has a fraction with a variable in the denominator.
• Excluding values from the numerator. Correction: numerator zero is usually allowed; denominator zero is not allowed. Warning sign: you set every expression in the fraction not equal to zero.
• Assuming the range is always all real numbers. Correction: check which outputs the rule can actually produce. Warning sign: the domain is finite, restricted, or the rule is a constant or fraction.
• Treating interval endpoints carelessly. Correction: use $\le$ when an endpoint is included and $<$ when it is excluded. Warning sign: the question uses phrases such as "up to but not including."
• Filling gaps in a piecewise range. Correction: list only outputs that are actually produced by the branches. Warning sign: one branch gives a single constant output separated from an interval.

Practice Tasks

Foundation

1. Find the domain and range of $\{(2,5),(3,7),(4,7),(5,9)\}$.
2. In the table below, state the domain and range.

| $x$ | $y$ | | ---: | ---: | | $-1$ | $4$ | | $0$ | $2$ | | $1$ | $2$ | | $2$ | $6$ |

1. Explain why repeated outputs are written once in the range.

Skill-Building

1. Make a table for $f(x)=2x-1$ using $x=-1,0,1,2$, then state the domain and range.
2. Find the domain of $g(x)=\frac{4}{x+3}$.
3. Find the domain of $h(x)=\frac{x+1}{x^2-9}$.
4. For $f(x)=x^2$ with domain $\{-2,-1,0,1,2\}$, state the range.

Exam-Style

1. A function is defined by $p(x)=x+2$ for $-3 \le x < 1$ and $p(x)=5$ for $1 \le x \le 4$. State the domain and range.
2. Find the domain of $r(x)=\frac{2x-1}{x^2+x-6}$, showing the denominator restriction.
3. For $q(x)=3x-4$ with domain $\{0,1,2,3\}$, state the range and write the corresponding ordered pairs.

Challenge

1. Explain why the domain and range of an inverse relation are swapped.
2. Give an example of a function with domain $\{1,2,3\}$ and range $\{4,9\}$. Explain why the range has fewer elements.
3. Create a piecewise function whose domain is $0 \le x \le 6$ but whose range has a gap. State both domain and range.

Generated Question Layer

• Direct set questions: find domain and range from ordered pairs or tables.
• Restriction questions: identify values excluded by denominators.
• Function-rule questions: produce outputs from a stated finite domain.
• Piecewise questions: combine intervals and output sets.
• Inverse-readiness questions: swap domain and range and explain the result.

Learner Aid Opportunities

• chart: Build an input-output organizer that separates "domain: allowed before using the rule" from "range: produced after using the rule."
• graph: Prepare a graph-reading overlay that marks domain as horizontal coverage and range as vertical coverage, with endpoint markers planned but not embedded here.
• interactive: Design a table-to-set task where learners remove repeated outputs and then compare their set with automated feedback.
• LLM tutor: Use a hint sequence that asks for denominator restrictions first, then table outputs, then final domain/range notation.

Exam-Derived Signals

These signals are assessment leads, not verified official past-question links. They should be checked against original papers and marking schemes before being used as final learner-facing references.

| Source | Current Signal | Review Status | Use Carefully As | | --- | --- | --- | --- | | data/exam_format_topic_crosswalk_2022.jsonl | Official 2022 format group Linear Programming/Functions/Relations maps to this topic and sibling pages; one item; weight 7.14. | Official format mapping; topic-page use still unreviewed. | Evidence that domain and range belong to the functions/relations assessment area. | | data/topic_frequency_2021_2025.json | topic-domain-and-range has total count 5 in the 2021-2025 extraction set. | Unreviewed aggregate. | Rough retrieval priority, not a final frequency claim. | | data/question_map_2021_2025.jsonl | Examples include a 2022 rational-function domain/range item, a 2023 piecewise-function domain/range item, and a 2024 inverse-domain/range item. | mapped_unreviewed. | Leads for future reviewed past-question linking. | | data/question_map_2021_2025.jsonl | Some records appear noisy, including fraction-ordering items mapped to this topic. | Needs human review. | Warning that automatic mapping may overstate coverage. |

Source And Review Notes

• Topic registry status: official in data/curriculum_map.json.
• Learner expansion status: original prose drafted from the official syllabus topic and local assessment signals.
• Exam mapping status: unreviewed except for the official exam-format crosswalk group.
• Review risk: extracted question mappings include likely false positives, so no past question should be treated as reviewed on this page yet.