Domain and range: what can go in and what can come out

Explain it like I am five

Picture a vending machine with a tray for the choices it accepts and a tray for the things it delivers. You choose something that can go in. The machine follows its rule and gives you one result.

A function works the same way. Its domain is the whole collection of allowed inputs. Its range is the whole collection of outputs it actually makes. You keep these collections separate because choosing a button and receiving a snack are different jobs.

You may have several buttons that deliver the same snack. List that snack once in the output collection. A choice the machine cannot accept belongs in neither its input tray nor its domain.

The input tray supplies the domain, and the results that reach the output tray form the range.

Reminder

Ordered pairs. The input comes first: (4, 11) records input 4 and output 11.
Squaring a negative. (−2 $) squared$ = (−2) × (−2) = 4, so negative and positive inputs may share an output.

Why it works. An ordered pair records an input first and its output second. Collecting the first coordinates tells you exactly which choices were used, so it gives the domain. Collecting the second coordinates gives the results, so it gives the range. These are sets, meaning collections in which listing an element twice adds nothing. A formula may accept a number arithmetically while a real situation rejects it. You can square a negative ticket count, but a negative number of tickets does not describe a purchase.

RuleDomain = all permitted inputs.
Range = all outputs actually reached by those inputs; a function assigns one output to each input.

The same idea, five ways

Say it

Say domain as allowed inputs and range as reached outputs.

Write it

The domain collects every permitted input; the range collects every output actually produced.

In math

For {(−4, 6), (0, 6), (3, −2), (5, 9)}:
Domain = {−4, 0, 3, 5}
Range = {−2, 6, 9}
Graph words: input positions and output heights

The accepted choices and delivered snacks of a vending machine.

See it

The input tray supplies the domain, and the results that reach the output tray form the range.

The same idea, other ways

As two trays

The domain tray holds every choice the machine accepts. The range tray holds every result it can deliver. A result must actually come from an allowed choice.

Four allowed inputs can produce only three different outputs.

With small numbers

For $x^{2}$ with allowed inputs {−2, 0, 2}, the domain is {−2, 0, 2} and the range is {0, 4}. Both −2 and 2 produce 4, and sets list 4 once.

As a question

Ask, What may I put in? for the domain. Ask, What can actually come out? for the range. Do not answer the second question by listing the inputs again.

In a real situation

An independent variable is the input you select or record. A dependent variable is the resulting output. A movie title can be the input and its earnings the output, so the two sets need not even contain the same kind of object.

Record	Input, or independent variable	Output, or dependent variable
Movie earnings	Movie title	Gross receipts in dollars
Ticket sales by year	Year in the recorded data	Tickets sold that year
A formula y = $x^{2}$	Chosen real number x	Calculated number y

.1Know cold

Keep a few decisions ready without a reference. These decisions tell you which arithmetic checks to make and which boundary symbols to use. The reminders below are memory devices, not substitutes for understanding the reasons you will learn.

Domain is input; range is output. Memory device: in goes with input and domain, out with output and range. (4, 11) means input 4 produces output 11.
A denominator cannot equal zero. Memory device: zero below means no go. $5 over 0$ cannot produce a quotient because no number times 0 equals 5. Developed in Find the domain by checking the arithmetic.
An even-root radicand must be nonnegative. Memory device: even root, zero or more. $square root of 0$ = 0 because $0 squared$ = 0. Developed in Find the domain by checking the arithmetic.
A bracket includes an endpoint; a parenthesis leaves it out. Memory device: square brackets hold the endpoint. [2, 5) includes 2 and excludes 5. Developed in Write the same set three ways.
Negative multiplication or division reverses an inequality. Memory device: a negative turns the number line around. −2x < 10 becomes x > −5 because division by −2 reverses the order. Taught in the inequality refresher and used in Find the domain by checking the arithmetic.
Choose a piecewise condition before its formula. Memory device: choose the lane, then calculate. For a rule giving 7 if x ≤ 2 and 9 if x > 2, input 2 gives 7 because the first condition includes equality. Developed in Write and evaluate a piecewise function.

Domain: what goes in

Range: what comes out

Choose the condition before calculating

These short reminders name the decisions you need quickly.

Reminder

Ordered pairs. The input comes first: (4, 11) records input 4 and output 11.
Squaring a negative. (−2 $) squared$ = (−2) × (−2) = 4, so negative and positive inputs may share an output.

The same idea, five ways

Say it

Remember the decisions before calculating.

Write it

Domain concerns inputs; range concerns outputs.

In math

(4, 11): input 4, output 11
Denominator ≠ 0
Even-root radicand ≥ 0
[2, 5) = {x | 2 ≤ x < 5}

Keep a few road signs recognizable so you can make the next decision quickly.

See it

Domain: what goes in

Range: what comes out

Choose the condition before calculating

These short reminders name the decisions you need quickly.

Worked exampleUse the input-output reminder

A function takes 4 as input and produces 11. Which collection contains 4, and which contains 11?

Input 4 goes into the function; output 11 comes out.

We need to identify which collection contains the input 4 and which contains the output 11.The question asks us to distinguish the role of an input from the role of its output.
Place 4 in the domain.4 was the allowed input.
Place 11 in the range.11 was an output the function produced.

Answer

4 belongs to the domain.
11 belongs to the range.

Check The record (4, 11) puts the domain member first and the range member second.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Use the same comparison after negative division: −2x < 10 becomes x < −5.

Negative division reverses order; x = 0 satisfies the original but not the proposed wrong result.

✓ Instead: x > −5.

✗ Not this: Treat every root equal to zero as undefined.

Zero is a permitted root output; division by that output would be a separate operation.

✓ Instead: = 0, but is undefined.

✗ Not this: Keep the endpoint 5 in [2, 5).

The right parenthesis records exclusion.

✓ Instead: 2 ≤ x < 5.

✗ Not this: Choose the condition x > 2 when the input is 2.

Strictly greater excludes equality.

✓ Instead: Use the condition x ≤ 2, giving the assigned output 7 in the tiny example.

Tips and tricks

Use the reminders to choose a check, then use the developed lesson for the reason and full worked method.

.2Understand, then rebuild it when needed

Do not memorize a separate answer for every shifted formula. Learn which operations can fail and how a graph covers its axes. Then rebuild a domain or range from the actual formula. A remembered answer from a different function can move the boundary to the wrong place.

Rebuild a denominator exclusion by setting the denominator equal to zero.
Rebuild an even-root restriction by solving a nonnegative-radicand inequality.
Rebuild the toolkit ranges from distance, squaring, cubing and division.
Rebuild a changed function's range by asking which outputs can be produced.
Rebuild a piecewise graph one interval at a time. Read its shadows after all pieces are present.

Squaring the input explains the output without a memorized table.

Reminder

Ordered pairs. The input comes first: (4, 11) records input 4 and output 11.
Squaring a negative. (−2 $) squared$ = (−2) × (−2) = 4, so negative and positive inputs may share an output.

The same idea, five ways

Say it

Rebuild the result from the formula’s operations.

Write it

A changed formula needs its own arithmetic checks.

In math

x + 4 ≥ 0
x ≥ −4
{x | x ≥ −4}
[−4, ∞)

Follow the current recipe rather than copying the answer from a different recipe.

See it

Rebuild the boundary by checking the inside: input −4 makes x + 4 equal zero, and its principal root is zero.

Worked exampleRebuild rather than guess

For the allowed input −2, what does the rule square the input produce?

The squared input produces 4 because two negative factors give a positive product.

We need the output produced by squaring the allowed input −2.The question gives the input and the rule, so our task is to apply the rule to that input.
Multiply −2 by itself: (−2 $) squared$ = (−2) × (−2) = 4.Squaring means multiplying the number by itself, and two negative factors give a positive product.

Answer

The output is 4.

Check The input 2 also gives 4, so the output does not reveal which sign went in.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Use an old function's remembered endpoint for a new formula.

The actual denominator or radicand may become zero at a different input.

✓ Instead: Rebuild the condition from the current expression.

Tips and tricks

Write the failing operation before trying to remember an answer.

.3Put on the cheat sheet

Use a small practice reference for details that are quicker to reconstruct than to memorize. The exam is closed book, so the sheet helps you study rather than giving you permission to bring it. Practice explaining each entry and rebuilding an answer after you cover the sheet.

Notation key: < or > uses a parenthesis; ≤ or ≥ uses a bracket; infinity always uses a parenthesis.
Domain checklist: context, every denominator, every even-root radicand, then keep all restrictions at once.
Toolkit comparison table: formula, domain and range for the nine basic functions.
Piecewise checklist: boundaries, condition, formula, endpoint dots, domain union, then range union.
Range check: show why forbidden outputs fail and why every claimed output can occur.

Domain: check the arithmetic

Range: check the possible results

Endpoint: substitute it to decide

A compact reference keeps the method visible while you practice.

Reminder

Ordered pairs. The input comes first: (4, 11) records input 4 and output 11.
Squaring a negative. (−2 $) squared$ = (−2) × (−2) = 4, so negative and positive inputs may share an output.

The same idea, five ways

Say it

Read the reference, then practice with it covered.

Write it

A study reference records conventions and methods you can rebuild for the exam.

In math

{2, 7}: two inputs
{5}: one output
(−∞, ∞): all real inputs

A map legend reminds you what the signs mean while you practice reading a map.

See it

Domain: check the arithmetic

Range: check the possible results

Endpoint: substitute it to decide

A compact reference keeps the method visible while you practice.

Worked exampleA one-line reference example

A rule accepts exactly the inputs 2 and 7 and produces 5 for each. Write its domain and range.

Two allowed inputs share the single range member 5.

We need every allowed input and every output the rule actually reaches.State what is being found before choosing the calculation.
List {2, 7} for the domain.The allowed inputs are given as two separate choices.
List {5} for the range.Both choices produce the same output, so there is one range member.

Answer

Domain: {2, 7}.
Range: {5}.

Check There are two arrows going in and one distinct destination coming out; an interval [2, 7] would add choices the rule never offered.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Copy a range from the reference without identifying the function.

The reference explains basic behavior, while a changed formula may move or reverse output bounds.

✓ Instead: Use the checklist to rebuild the range of the given formula.

Tips and tricks

Cover the reference after one example and repeat the method in your own words.

Strategy: step by step

1. Decide what the input represents and what the output represents.
2. For listed ordered pairs or a table, collect the inputs for the domain and the outputs for the range.
3. Remove repeated entries because sets record membership, not frequency.
4. If a context is given, keep only physically meaningful inputs. Do not fill the spaces between separate listed values.

Strategy

Separate domain and range

Is the function supplied as pairs or a table?

YesCollect listed inputs and outputs without duplicates.

NoUse the formula or graph method taught next.

↓

Does a real context limit the inputs?

YesKeep meaningful choices, such as whole-number ticket counts.

NoKeep the domain specified by the mathematical description.

1. Decide what the input represents and what the output represents.
2. For listed ordered pairs or a table, collect the inputs for the domain and the outputs for the range.
3. Remove repeated entries because sets record membership, not frequency.
4. If a context is given, keep only physically meaningful inputs. Do not fill the spaces between separate listed values.

Worked exampleSeparate the input and output trays

Find the domain and range of {(−4, 6), (0, 6), (3, −2), (5, 9)}.

Four allowed inputs can produce only three different outputs.

We need every allowed input and every output the rule actually reaches.State what is being found before choosing the calculation.
Read the first coordinate of each ordered pair: −4, 0, 3, 5.An ordered pair writes the input before the output.
Collect the inputs: {−4, 0, 3, 5}.Only these four inputs are listed; numbers between them were not supplied.
Read the second coordinates: 6, 6, −2, 9.The second coordinate tells what came out for that input.
Collect the outputs without duplicates: {−2, 6, 9}.The two appearances of 6 refer to the same member of the range.

Answer

Domain: {−4, 0, 3, 5}.
Range: {−2, 6, 9}.

Check Draw four arrows. The arrows start at the four domain members and end at three different range members, with two arrows ending at 6.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: The listed domain {−4, 0, 3, 5} is the interval [−4, 5].

An interval adds every intervening real number, but the listed function supplied only four inputs.

✓ Instead: Keep the finite domain {−4, 0, 3, 5}.

✗ Not this: Two different inputs produce 6, so this is not a function.

A function requires one output per input. It permits several inputs to share an output.

✓ Instead: The pair list is a function, and 6 appears once in its range.

Tips and tricks

Write input and output above the two collections before listing their members.
Memory device: in identifies domain and out identifies range. Several inputs may share one output.
For a head count, list whole-number choices instead of filling in fractional people.

Trap. Replacing a listed domain {−4, 0, 3, 5} with [−4, 5]. That interval would add every number between the endpoints. A finite list includes only its listed entries.