PAPER · REASONING · PRACTICE

Functions: inputs, domains, tables and graphs

Decide whether a relation is a function, use three representations and check mathematical and contextual domains.

40 minutes · Year 8, lower volume · text pp. 49–58 · textbook lessons 25–29

Textbook reference: Year 8, lower volume · Chapter 19: Linear functions · lessons 25–29

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the lower-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

3 min: Predict whether repeated inputs or repeated outputs prevent a relation from being a function. 8 min: On paper compare tables {−1→1,0→0,1→1}, {0→0,0→1}, and {0→0,0→0,1→1}. Then model a tank filled at 3 L/min up to 60 L; test times 0,20,21 minutes. 5 min: A repeated identical pair adds no second output; conflicting outputs for the same input do. Explain why the filling relation applies only for 0≤t≤20 even though its algebraic expression can be evaluated later. 2 min: Individually give a function with two distinct inputs sharing one output, and explain why it is valid.

Organise the key idea

Every allowed input has one output. Different inputs can have the same output. A contextual domain may be smaller than the formula’s real domain.

Open the original chapter

Use the lesson question here for paper discussion; the original activity opens separately.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

Does this table define y as a function of x? Explain using each input.

x	y
-1	1
0	0
1	1
2	4

Use the stated conditions and the labelled scales.

Read each ordered pair as input, output.
One output per input is required.
Repeated output values are allowed.

Check repeated inputs, not merely repeated outputs.
No input has conflicting outputs.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

An initially empty 60 L tank fills at 3 L/min, stopping when full. Write V in terms of t while filling and state the time domain.

During filling, $V=\square t$ L, with t measured in minutes.
At capacity, $3t=60$ , so $t=\frac{60}{\square}=\square$ minutes.
Time domain: $0\leq t\leq\square$ ; explain both endpoints.

The tank starts empty, so volume is the constant rate multiplied by time, with no initial-volume term.
The restrictions are t≥0 and 3t≤60. The capacity determines when filling stops.
Calculate 60÷3 for the upper time limit. Include both the empty start and the just-full moment.

Volume=rate×time, and 60/3=20 minutes fills the tank.
Include the initially empty and just-full endpoints.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

For the given relation, find y when $x=3$ .

y=2x+1

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Substitute $x=3$ into the complete relation.
$2\times(3)+(1)=7$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

The relation contains only the pair $(2,-1)$ . On its stated input domain {2}, is it a function?

x	y
2	-1

Use the stated conditions and the labelled scales.

Read each ordered pair as input, output.
One output per input is required.
Repeated output values are allowed.

Check repeated inputs, not merely repeated outputs.
No input has conflicting outputs.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

Does this table define y as a function of x? Explain using each input.

x	y
0	1
0	2
1	3

Use the stated conditions and the labelled scales.

Read each ordered pair as input, output.
One output per input is required.
Repeated output values are allowed.

Check repeated inputs, not merely repeated outputs.
The repeated input has conflicting outputs.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

For the given relation, find y when $x=-1$ .

y=3x-2

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Substitute $x=-1$ into the complete relation.
$3\times(-1)+(-2)=-5$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

Read the x-coordinate where the shown graph meets the x-axis. Check it in the relation.

Use the stated conditions and the labelled scales.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

At the x-axis the output is zero.
0=−x+2 gives x=2.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

State the real input that must be excluded from this relation.

y=\frac{1}{x-2}

Inspect the original denominator.
Solve x−2=0.
Exclude that value.

The denominator cannot be zero.
x−2=0 at x=2.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Does this table define y as a function of x? Explain using each input.

x	y
1	5
2	5
3	5

Use the stated conditions and the labelled scales.

Read each ordered pair as input, output.
One output per input is required.
Repeated output values are allowed.

Check repeated inputs, not merely repeated outputs.
No input has conflicting outputs.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

A ticket costs 4 dollars. At most 20 tickets are available. State the possible ticket counts n and write the cost C.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Counts are whole numbers, including buying none.
The maximum count is 20; multiply the count by 4.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

For the given relation, find y when $x=0$ .

y=2x-2

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Substitute $x=0$ into the complete relation.
$2\times(0)+(-2)=-2$ .

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

Every input in the table has output 4. Is this a function? Is it a constant function?

x	y
-1	4
0	4
1	4

Use the stated conditions and the labelled scales.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Unique outputs per input do not have to differ between inputs.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

For $y=x^{2}$ , complete the outputs at $x=-2,0,2$ . Is one output shared by two inputs allowed?

Keep the minus sign inside the squared input.
Compute (−2)², 0² and 2².
Compare inputs rather than requiring every output to differ.

Square each signed input separately.
Both −2 and 2 give 4, without creating conflicting outputs for one input.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

Draw the graph of $y=x+1$ for $-2\leq x\leq2$ . Make a value table first.

Use the stated conditions and the labelled scales.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Substitute at least two distinct inputs and plot the ordered pairs.
Join the collinear points only over the stated input interval.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

The vertical line $x=2$ contains the points shown. Does it define y as a function of x?

x	y
2	-1
2	1

Use the stated conditions and the labelled scales.

Read each ordered pair as input, output.
One output per input is required.
Repeated output values are allowed.

Check repeated inputs, not merely repeated outputs.
The repeated input has conflicting outputs.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

The 60 L tank fills at 3 L/min and stops when full. A learner uses V=3t at t=21 to predict 63 L. Explain why the calculation does not give a valid filling-stage prediction.

Read the variables, axes and domain before using a formula.
Substitute known coordinates or identify the intercept or common value that the task requests.
Check the result in the original relation and retain any endpoint condition.

Find the fill time 60/3=20 minutes.
The model V=3t describes the filling stage only.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Preparation and assessment notes

Prerequisites

Coordinates, substitution and simple inequalities.

Look for in students’ work

Use the two prerequisite checks to choose support. Project the worked and guided tasks, then collect the eight paper core questions. Follow-ups and extensions are optional; accept other complete valid methods. This page does not save learners’ paper answers.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Original chapter-core practice after the cited lower-volume textbook lessons. References identify skills; textbook exercises are not reproduced.

大湾区初中数学三维课堂（八年级下册）正文.pdf

Year 8, lower volume · text pp. 49–58 · textbook lessons 25–29

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Functions: inputs, domains, tables and graphs

Year 8, lower volume · text pp. 49–58 · textbook lessons 25–29 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Does this table define y as a function of x? Explain using each input.

x	y
-1	1
0	0
1	1
2	4

Use the stated conditions and the labelled scales.

Guided attempt2

An initially empty 60 L tank fills at 3 L/min, stopping when full. Write V in terms of t while filling and state the time domain.

During filling, $V=\square t$ L, with t measured in minutes.
At capacity, $3t=60$ , so $t=\frac{60}{\square}=\square$ minutes.
Time domain: $0\leq t\leq\square$ ; explain both endpoints.

Prerequisite3

For the given relation, find y when $x=3$ .

y=2x+1

Prerequisite4

The relation contains only the pair $(2,-1)$ . On its stated input domain {2}, is it a function?

x	y
2	-1

Use the stated conditions and the labelled scales.

Functions: inputs, domains, tables and graphs

Year 8, lower volume · text pp. 49–58 · textbook lessons 25–29 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Does this table define y as a function of x? Explain using each input.

x	y
0	1
0	2
1	3

Use the stated conditions and the labelled scales.

Independent practice6

For the given relation, find y when $x=-1$ .

y=3x-2

Independent practice7

Read the x-coordinate where the shown graph meets the x-axis. Check it in the relation.

Use the stated conditions and the labelled scales.

Independent practice8

State the real input that must be excluded from this relation.

y=\frac{1}{x-2}

Exit question9

Does this table define y as a function of x? Explain using each input.

x	y
1	5
2	5
3	5

Use the stated conditions and the labelled scales.

Exit question10

A ticket costs 4 dollars. At most 20 tickets are available. State the possible ticket counts n and write the cost C.

Functions: inputs, domains, tables and graphs — Further practice

Year 8, lower volume · text pp. 49–58 · textbook lessons 25–29

Further practice11

For the given relation, find y when $x=0$ .

y=2x-2

Further practice12

Every input in the table has output 4. Is this a function? Is it a constant function?

x	y
-1	4
0	4
1	4

Use the stated conditions and the labelled scales.

Further practice13

For $y=x^{2}$ , complete the outputs at $x=-2,0,2$ . Is one output shared by two inputs allowed?

Further practice14

Draw the graph of $y=x+1$ for $-2\leq x\leq2$ . Make a value table first.

Use the stated conditions and the labelled scales.

Extension15

The vertical line $x=2$ contains the points shown. Does it define y as a function of x?

x	y
2	-1
2	1

Use the stated conditions and the labelled scales.

Extension16

The 60 L tank fills at 3 L/min and stops when full. A learner uses V=3t at t=21 to predict 63 L. Explain why the calculation does not give a valid filling-stage prediction.

Functions: inputs, domains, tables and graphs — Teacher answers

Year 8, lower volume · text pp. 49–58 · textbook lessons 25–29

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Coordinates, substitution and simple inequalities.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

3′ Predict whether repeated inputs or repeated outputs prevent a relation from being a function.
8′ On paper compare tables {−1→1,0→0,1→1}, {0→0,0→1}, and {0→0,0→0,1→1}. Then model a tank filled at 3 L/min up to 60 L; test times 0,20,21 minutes.
5′ A repeated identical pair adds no second output; conflicting outputs for the same input do. Explain why the filling relation applies only for 0≤t≤20 even though its algebraic expression can be evaluated later.
2′ Individually give a function with two distinct inputs sharing one output, and explain why it is valid.

Worked example1

Does this table define y as a function of x? Explain using each input.

x	y
-1	1
0	0
1	1
2	4

Use the stated conditions and the labelled scales.

Reference answer

Yes. Every input has exactly one output; different inputs may share an output.

Check repeated inputs, not merely repeated outputs.
No input has conflicting outputs.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

An initially empty 60 L tank fills at 3 L/min, stopping when full. Write V in terms of t while filling and state the time domain.

During filling, $V=\square t$ L, with t measured in minutes.
At capacity, $3t=60$ , so $t=\frac{60}{\square}=\square$ minutes.
Time domain: $0\leq t\leq\square$ ; explain both endpoints.

Reference answer

$V=3t,\quad0\leq t\leq20$ .

V=3t,\quad0\leq t\leq20

Volume=rate×time, and 60/3=20 minutes fills the tank.
Include the initially empty and just-full endpoints.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

For the given relation, find y when $x=3$ .

y=2x+1

Reference answer

$y=7$

y=7

Substitute $x=3$ into the complete relation.
$2\times(3)+(1)=7$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

The relation contains only the pair $(2,-1)$ . On its stated input domain {2}, is it a function?

x	y
2	-1

Use the stated conditions and the labelled scales.

Reference answer

Yes. Every input has exactly one output; different inputs may share an output.

Check repeated inputs, not merely repeated outputs.
No input has conflicting outputs.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

Does this table define y as a function of x? Explain using each input.

x	y
0	1
0	2
1	3

Use the stated conditions and the labelled scales.

Reference answer

No. At least one input is paired with two different outputs.

Check repeated inputs, not merely repeated outputs.
The repeated input has conflicting outputs.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

For the given relation, find y when $x=-1$ .

y=3x-2

Reference answer

$y=-5$

y=-5

Substitute $x=-1$ into the complete relation.
$3\times(-1)+(-2)=-5$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

Read the x-coordinate where the shown graph meets the x-axis. Check it in the relation.

Use the stated conditions and the labelled scales.

Reference answer

x=2, since y=0 there.

x=2

At the x-axis the output is zero.
0=−x+2 gives x=2.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Use the stated conditions and the labelled scales.

Independent practice8

State the real input that must be excluded from this relation.

y=\frac{1}{x-2}

Reference answer

$x\neq2$

x\neq2

The denominator cannot be zero.
x−2=0 at x=2.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Does this table define y as a function of x? Explain using each input.

x	y
1	5
2	5
3	5

Use the stated conditions and the labelled scales.

Reference answer

Yes. Every input has exactly one output; different inputs may share an output.

Check repeated inputs, not merely repeated outputs.
No input has conflicting outputs.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

A ticket costs 4 dollars. At most 20 tickets are available. State the possible ticket counts n and write the cost C.

Reference answer

$C=4n,\quad n\in\{0,1,\ldots,20\}$

C=4n,\quad n\in\{0,1,\ldots,20\}

Counts are whole numbers, including buying none.
The maximum count is 20; multiply the count by 4.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

For the given relation, find y when $x=0$ .

y=2x-2

Reference answer

$y=-2$

y=-2

Substitute $x=0$ into the complete relation.
$2\times(0)+(-2)=-2$ .

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

Every input in the table has output 4. Is this a function? Is it a constant function?

x	y
-1	4
0	4
1	4

Use the stated conditions and the labelled scales.

Reference answer

Yes to both: y=4 gives one output for every input.

Unique outputs per input do not have to differ between inputs.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

For $y=x^{2}$ , complete the outputs at $x=-2,0,2$ . Is one output shared by two inputs allowed?

Reference answer

Outputs 4,0,4; yes, repeated outputs are allowed.

Square each signed input separately.
Both −2 and 2 give 4, without creating conflicting outputs for one input.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

Draw the graph of $y=x+1$ for $-2\leq x\leq2$ . Make a value table first.

Use the stated conditions and the labelled scales.

Reference answer

Use points (−2,−1),(0,1),(2,3), then join them on the stated domain.

Substitute at least two distinct inputs and plot the ordered pairs.
Join the collinear points only over the stated input interval.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Use the stated conditions and the labelled scales.

Extension15

The vertical line $x=2$ contains the points shown. Does it define y as a function of x?

x	y
2	-1
2	1

Use the stated conditions and the labelled scales.

Reference answer

No. At least one input is paired with two different outputs.

Check repeated inputs, not merely repeated outputs.
The repeated input has conflicting outputs.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

The 60 L tank fills at 3 L/min and stops when full. A learner uses V=3t at t=21 to predict 63 L. Explain why the calculation does not give a valid filling-stage prediction.

Reference answer

21 minutes is outside 0≤t≤20. The formula gives 63 algebraically, but the stopped tank holds 60 L.

Find the fill time 60/3=20 minutes.
The model V=3t describes the filling stage only.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.