A function gives one answer to each input

Explain it like I am five

Picture a vending machine with a fixed menu. You choose a button. That choice is the input. The snack it gives you is the output. Any collection of these matched choices and results is a relation, like a list of (button, snack) pairs. A function is a relation that gives exactly one answer for each allowed choice.

Two buttons may both give crackers. Each button still has one answer. Trouble begins if one button is assigned both crackers and pretzels, with no instruction choosing between them.

The domain lists the allowed inputs. The range lists only outputs that actually appear. You will read the same matched pairs in lists, tables, pictures, and formulas.

An allowed button determines one snack.

Reminder

Ordered pairs and sets. (1, 2) records input 1 followed by output 2. A set such as {2, 4} lists distinct values.

Why it works. A relationship is useful as a function because knowing an input settles the output. If one input had two different assigned outputs, a request for its value would have no definite answer. Repeated outputs do not cause that problem: you still know what happens at each input. You must choose the direction first. A menu can identify a price from an item even when a price cannot identify a unique item.

RuleA relation is a set of ordered pairs (input, output). It is a function when every input in its domain has exactly one output. The range is the set of outputs it produces.

The same idea, five ways

Say it

y is a function of x.

Write it

Each allowed input has exactly one output. In f(x) = 2x, f names the doubling rule, x is the input, and f(x), read f of x, is its output.

In math

{(1, 2), (2, 4), (3, 6)}
y = 2x
f(x) = 2x
On a graph, each allowed horizontal input has one height.

One accepted vending-machine button selects one snack.

See it

Each input sends one arrow to its output.

The same idea, other ways

As a machine

One allowed button must settle which snack the rule gives. A missing choice is outside the domain.

A gives a definite output.

As arrows

Many arrows may arrive at the same output. Each input must send exactly one arrow.

Different inputs may share crackers.

As a question

Start with an item and ask its price. If the menu has one price per item, your question has a definite answer.

With numbers

The rule $x^{2}$ gives 4 at both 2 and −2. Each individual input still has only one square.

Given as	How to check
Ordered pairs	No input has two different paired outputs.
A table	Compare all columns sharing the same input.
A graph	Slide an upright ruler across the graph. Two contacts at the same horizontal position mean one input has two outputs. The line-test lesson explains this picture.
An equation	Choose an input and find every permitted output. For $x^{2}$ + $y^{2}$ = 1 at x = 0, y = 1 and y = −1 both work, so this relation fails.

.1Domain and range

The domain is the menu of accepted inputs. The range is the collection of answers actually reached. A relation has a domain and range even if it fails to be a function. Read each pair as input first, output second. The range lists only outputs that actually appear.

Domain: first coordinates of the ordered pairs.
Range: second coordinates of the ordered pairs, without duplicate set members.
Inputs may be numbers or labels. Natural numbers are 1, 2, 3 and so on.

The two arrows arriving at 4 are allowed.

The same idea, five ways

Say it

Domain means what may go in. Range means what comes out.

Write it

List each distinct input once for domain and each distinct output once for range.

In math

{(−2, 4), (0, 0), (2, 4)}
Domain: {−2, 0, 2}
Range: {0, 4}

The available buttons and the available snacks are different lists.

See it

The two arrows arriving at 4 are allowed.

Worked exampleShared outputs do not add extra range members

For the relation {(−2, 4), (0, 0), (2, 4)}, decide function status and list its domain and range.

The two arrows arriving at 4 are allowed.

What it asks. Decide whether the three listed inputs each have one output, then list domain and range.

Plan. Read every first coordinate, then every second coordinate. Remove repeated set members, then check the input arrows.

The first numbers are −2, 0, and 2.
The output 4 appears twice, but it is one range member.

Read the three first coordinates: −2, 0 and 2.These are the accepted inputs.
Read the second coordinates: 4, 0 and 4, then list the distinct values 0 and 4.Range is a set, so a repeated output is listed once.
Check each input's arrows. Each input points to only one output.Different inputs may share the output 4 without creating ambiguity at either input.

Answer

Function.
Domain: {−2, 0, 2}.
Range: {0, 4}.

Check This is the rule 'square the input' restricted to the three listed inputs, and each square matches its pair.

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

✗ Not this: The range of {(1, 2), (3, 2)} has two distinct members because the output 2 appears twice.

Two occurrences of the same value do not make two different set members.

✓ Instead: The range has one member: {2}.

Tips and tricks

Read first coordinates for domain and second coordinates for range.
As a guest list: The domain names who may enter; the range names results that actually occur.
As two collections: In {(1, 2), (3, 2)}, the domain has two members and the range has one.

.2Odd and even labels

Odd and even sort whole counting numbers into two labels. Even numbers split into pairs with nothing left over; odd numbers leave one unpaired object. A label can describe many numbers, so starting with the label does not choose one particular number.

2 and 4 are even because 2 = 2 × 1 and 4 = 2 × 2.
1, 3 and 5 are odd because each leaves one after pairing.
Number to label, odd or even, is a function for these inputs. Label to number is not, because one label names several numbers.

One label has arrows to several different numbers.

The same idea, five ways

Say it

An even number makes complete pairs. An odd number leaves one over.

Write it

Starting with a number tells you its label; starting with the label does not identify one number.

In math

2 → even
4 → even
1 → Odd
3 → Odd
Odd → 1 and Odd → 3

Four chairs form two pairs. Three chairs leave one chair unpaired.

See it

One label has arrows to several different numbers.

Worked exampleOdd and even labels do not determine a unique number

In the pictured relation, each label points to natural numbers of that type. Does the label determine a number as a function?

One label has arrows to several different numbers.

What it asks. Starting with Odd or even, can you recover one number?

Plan. Read every arrow leaving each label. If a label has several different destinations, label to number is not a function.

Odd is the starting label in three of the arrows.
A label input must obey the same one-output condition as a number input.

Odd means a counting number not divisible into whole-number pairs; even means one divisible into such pairs.The input labels need meanings before their outputs can be interpreted.
Input Odd points to 1, 3 and 5.Each of those listed natural numbers is odd.
Input even points to 2 and 4.Each of those listed natural numbers is even.
Reject function status for the direction label to number.Each input label is assigned several different outputs.

Answer

Not a function from label to number.

Check Reverse the listed pairs. Every number from 1 through 5 has one label, so number to label is a function. Changing direction changes the question.

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

✗ Not this: Odd identifies one number because it is one word.

One input word can still have several assigned outputs.

✓ Instead: Odd maps to 1, 3 and 5 in this relation, so it does not determine a unique output.

Tips and tricks

Labels can be inputs; test their assigned outputs the same way you test numeric inputs.
As pairing: Four chairs form two pairs. Five chairs form two pairs and one leftover chair.
As grouped labels: The word Odd describes several choices, so it cannot tell you which number was chosen.

.3Choose the direction

You decide which row is the starting row. A price list can answer 'What does tea cost?' even when it cannot answer 'What item costs three dollars?' Directions are part of the question, not a feature you can ignore.

The independent variable is the input you choose from the domain.
The dependent variable is the output determined by that input.

Tea and juice have one price each but share the output 3.

The same idea, five ways

Say it

Price is a function of item.

Write it

Start with the item to find its price. Reversing the direction asks a different question.

In math

Tea → 3
Juice → 3
3 → Tea and 3 → Juice

Two snacks can have the same price, so the receipt alone may not name the snack.

See it

Tea and juice have one price each but share the output 3.

Worked exampleRead a menu in both directions

The pictured menu assigns a price to each item. Is price a function of item? Is item a function of price?

Tea and juice have one price each but share the output 3.

What it asks. Check item to price, then check price to item.

Plan. Choose the input row first. Follow each column to its partner. Reverse the rows and check again.

Tea and Juice both give 3 dollars.
When 3 dollars becomes the input, count all the item outputs.

Read the columns from item to price: tea gives 3, cocoa gives 4, and juice gives 3.Each item has a single listed price.
Reverse the direction. Input price 3 gives both tea and juice.The reversed relationship has one input with two different outputs.
Conclude that item-to-price is a function and price-to-item is not.Function status depends on which quantity you chose as input.

Answer

Price is a function of item.
Item is not a function of price.

Check Two items sharing a price causes no uncertainty when you start with an item. It creates uncertainty when you start with price 3.

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

✗ Not this: A menu is a function in both directions because every item has a price.

Shared prices create several outputs if price becomes the input.

✓ Instead: Check each chosen direction separately.

Tips and tricks

Write 'input → output' in words before deciding function status.
As a menu: An item selects one price, while one price can name several items.
As a reversed lookup: Start with Tea and the answer is 3. Start with 3 and both Tea and Juice match.

Strategy: step by step

Name what you are starting with; that quantity is the input.
Name what you want to find; that quantity is the output.
Inspect every input and its assigned outputs.
Accept shared outputs. Reject an input with two different outputs.
List distinct inputs as the domain and distinct outputs as the range.

Strategy

Check one output for each input

Are the pairs listed or arranged in a table?

YesFind repeated inputs and compare the outputs assigned to each. A matching repeated pair causes no conflict.

NoUse the graph or equation branch.

↓

Is one input assigned two different outputs?

YesWrite not a function, followed by that input and both outputs.

NoEvery listed input has one output, so the listed relation is a function.

↓

Are you given a graph?

YesSlide an upright ruler across it. If one horizontal input position has two graph heights, the relation is not a function.

NoFor an equation, choose an input and solve for every possible output. Two different outputs at one input disprove a function.

Name what you are starting with; that quantity is the input.
Name what you want to find; that quantity is the output.
Inspect every input and its assigned outputs.
Accept shared outputs. Reject an input with two different outputs.
List distinct inputs as the domain and distinct outputs as the range.

Worked exampleFunction or not from ordered pairs

Decide whether each relation gives one output per input.
(a) {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}
(b) {(1, 3), (2, 5), (1, 7)}

In (a), each input sends one arrow.

Two arrows leave input 1 for different outputs.

What it asks. For each list, does one input have two different outputs? Also list the first relation’s allowed inputs and actual outputs.

Plan. Read the first number in each pair as input. Compare pairs with the same input, then collect distinct first and second numbers.

In (1, 3), 1 is the input and 3 is its output.
In list (b), find both pairs whose first number is 1.

For (a), each of the inputs 1, 2, 3, 4 and 5 has one paired output.The first coordinate is the input, and none has conflicting outputs.
Collect the domain {1, 2, 3, 4, 5} and the range {2, 4, 6, 8, 10}.Domain lists accepted inputs; range lists produced outputs.
For (b), input 1 pairs with both 3 and 7.A function cannot give two different answers for the same input.

Answer

(a) function.
Domain: {1, 2, 3, 4, 5}.
Range: {2, 4, 6, 8, 10}.
(b) Not a function. Input 1 has outputs 3 and 7.

Check In (a), the instruction 'double the input' gives the listed output every time. In (b), asking for the output at 1 leaves two competing answers.

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

✗ Not this: Input 1 has output 3 and output 7, so choose whichever you prefer.

The relation itself must assign a definite answer; choosing later changes the problem.

✓ Instead: This relation is not a function.

✗ Not this: Two inputs share output 4, so it is not a function.

Neither input has conflicting outputs.

✓ Instead: It can be a function; shared outputs are allowed.

✗ Not this: Write the range of {(−2, 4), (0, 0), (2, 4)} as [0, 4].

That interval includes every number between 0 and 4, but the relation produces only 0 and 4.

✓ Instead: Write the separate outputs in braces: {0, 4}.

Tips and tricks

First identify the input. The phrase 'B is a function of A' means start with A and find B.
Memory cue: one arrow OUT of each input. Many arrows IN to one output are allowed.
On the exam, give the evidence: not a function, because input 1 has outputs 3 and 7.
A few separate values go in braces, such as {0, 4}. An interval means a whole stretch, such as [0, 4], which also contains 1 and 2.5.

Trap. Checking for repeated outputs instead of conflicting inputs. Two arrows arriving together are allowed; two arrows leaving one input for different answers are not.