Precalculus-I · Unit 1 · Video 1 · Interactive Practice

Relations and Functions: One Input, One Output

IKey Formulas

Term	Example	Meaning
Relation	$\{(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)\}$	A set of ordered pairs $(x, y)$
Domain	$\{1, 2, 3, 4, 5\}$	The first components: the inputs, or independent variable $x$
Range	$\{2, 4, 6, 8, 10\}$	The second components: the outputs, or dependent variable $y$
Function	$1 \to 2$ , $2 \to 4$ , …, $5 \to 10$	A relation in which each input is paired with exactly one output

Key Insight: Several inputs may share one output, but no input may have two. Whether a relationship is a function depends on which quantity you take as the input.

IIVisualization 1 — Count the Arrows Leaving Each Input

An arrow diagram is a function exactly when one arrow leaves every input in the domain.

💡 Challenge: build a function with domain $\{p, q, r\}$ and range $\{n\}$ , then one with the same domain that is not a function.

IIIVisualization 2 — Reverse Every Pair

Swapping inputs and outputs can turn a function into a non-function, or keep it a function.

Relation

IVVisualization 3 — Percent Grade and GPA

Every whole-number percent lies in exactly one band, but one GPA covers many percents.

Input Percent 82 GPA

VQuiz Questions

Problem 1 · Domain and Range

Given: the relation $\{(-3, 5), (0, 2), (4, 5), (6, -1)\}$ — find its domain and its range.

What is the domain?

What is the range?

✅ Correct! The domain collects the first components and the range the second components; the repeated

5

is listed once.

❌ Swapped. The domain is the set of first components (inputs); the range is the set of second components (outputs).

❌ Not quite. A relation given as a list of pairs has a domain and range made of the listed values themselves — every one of them, not just the smallest and largest.

❌ Not quite. That set mixes inputs and outputs. The domain uses only the first number of each pair, the range only the second.

❌ Check the domain. Read off the first number in each pair.

❌ Check the range. Read off the second number in each pair.

Show solution

The first components are $-3$ , $0$ , $4$ , $6$ , so the domain is $\{-3, 0, 4, 6\}$ .

The second components are $5$ , $2$ , $5$ , $-1$ . A set lists each element once, so the range is $\{-1, 2, 5\}$ .

The relation is also a function: each input has one output, and the inputs $-3$ and $4$ share the output $5$ , which is allowed.

Problem 2 · Shared Outputs

Which of these relations is not a function?

✅ Correct! The input

3

is paired with two outputs,

1

and

2

, so this relation is not a function.

❌ Not quite. Here several inputs share one output. That is allowed: the definition limits the outputs of each input, not how many inputs reach an output.

❌ Not quite. The inputs

2

4

6

are all different, and each has one output. Seeing

2

and

4

in both roles does not matter.

Show solution

For each relation, list the inputs and count the outputs of each one.

(a) $1 \to 3$ , $2 \to 3$ , $3 \to 3$ : one output each, a function (all share the output $3$ ).
(b) $-1 \to 0$ , $0 \to -1$ , $1 \to 0$ : one output each, a function.
(c) $3 \to 1$ and $3 \to 2$ : the input $3$ has two outputs, not a function.
(d) $2 \to 4$ , $4 \to 2$ , $6 \to 6$ : one output each, a function.

Problem 3 · Repair a Relation

Given: $R = \{(1, 4), (2, 5), (3, 4), (2, 6)\}$ — find the one pair whose removal makes $R$ a function, and the range of what is left.

Which pair should be removed?

What is the range after the removal?

✅ Correct! Without

(2, 6)

the relation is

\{(1, 4), (2, 5), (3, 4)\}

: each input has one output, and the range is

\{4, 5\}

❌ Not quite. The inputs

1

and

3

sharing the output

4

is allowed. The trouble is an input with two outputs.

❌ Not quite. Only the input

2

has two outputs, so removing one of its two pairs is enough.

❌ Check the pair. Find the input that appears in two pairs.

❌ That is the domain. The range is the set of second components.

❌ Not quite. The

6

left with the removed pair

(2, 6)

; no remaining pair has

6

as its output.

❌ Check the range. List the second component of each of the three remaining pairs, once each.

Show solution

The inputs and their outputs are $1 \to 4$ , $2 \to 5$ and $2 \to 6$ , $3 \to 4$ . Only the input $2$ has two outputs, so removing $(2, 6)$ (or $(2, 5)$ , which is not offered) makes $R$ a function.

R' = \{(1, 4), (2, 5), (3, 4)\}

The second components are $4$ , $5$ , $4$ , so the range is $\{4, 5\}$ . The domain is still $\{1, 2, 3\}$ .

Problem 4 · Which Way Is a Function?

Given: a cinema charges $8 for a child ticket, $12 for a student ticket, $12 for an adult ticket and $8 for a senior ticket. Decide both directions.

Is price a function of ticket type?

Is ticket type a function of price?

✅ Correct! The same four pairs give a function from ticket type to price and fail the test from price to ticket type.

❌ Not quite. With ticket type as the input, $12 is an output, and two inputs may share an output.

❌ Wrong direction. That treats the price as the input. In "price is a function of ticket type" the ticket type is the input.

❌ Not quite. The conclusion is right but the reason is false: $8 goes with child and senior.

❌ Wrong direction. That answers the first question. Here the price is the input.

❌ Not quite. Sharing is allowed for outputs, but here an input, $12, has two outputs.

❌ Not quite. Few inputs is no help if one input still has two outputs.

❌ Check the first direction. Take ticket type as the input.

❌ Check the second direction. Take price as the input.

Show solution

Ticket type as input: child $\to$ $8, student $\to$ $12, adult $\to$ $12, senior $\to$ $8. Each input has one output, so price is a function of ticket type. The shared prices are allowed.

Price as input: $8 $\to$ child and senior; $12 $\to$ student and adult. Each input has two outputs, so ticket type is not a function of price.

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