Worksheet: Function Rules

In this worksheet, we will practice finding a function rule from a given function table.

Q1:

Find the rule for the following function table.

Input (π‘₯) 1 2 3 4 5
Output (β‹―) 10 14 18 22 26
  • A 4 π‘₯ + 6
  • B 6 π‘₯ + 4
  • C 6 π‘₯ βˆ’ 4
  • D 4 π‘₯ βˆ’ 6
  • E π‘₯ + 6

Q2:

Find the rule for the given function table.

Input (π‘₯) 1 4 10
Output (β‹―) 9 12 18
  • A 8 π‘₯ βˆ’ 8
  • B 8 βˆ’ π‘₯
  • C π‘₯ + 8
  • D 8 π‘₯ + 8
  • E π‘₯ βˆ’ 8

Q3:

Find the function rule for this table. Then calculate the two missing numbers.

Input (π‘₯) 12 13 14 15 16
Output (𝑦) 76 82 88 ? ?
  • A 𝑦 = 7 π‘₯ + 5 , 94, 117
  • B 𝑦 = 7 π‘₯ + 5 , 110, 100
  • C 𝑦 = 6 π‘₯ + 4 , 94, 20
  • D 𝑦 = 6 π‘₯ + 4 , 19, 100
  • E 𝑦 = 6 π‘₯ + 4 , 94, 100

Q4:

By using the linear relation π‘Ž+2𝑏=βˆ’8, fill in the missing values in the table.

π‘Ž 2 β‹― βˆ’ 8
𝑏 β‹― βˆ’ 3 β‹―
  • A 𝑏 = βˆ’ 1 2 , π‘Ž = 5 2 , 𝑏 = 8
  • B 𝑏 = βˆ’ 5 , π‘Ž = βˆ’ 1 4 , 𝑏 = 0
  • C 𝑏 = βˆ’ 1 2 , π‘Ž = βˆ’ 5 2 , 𝑏 = 8
  • D 𝑏 = βˆ’ 5 , π‘Ž = βˆ’ 2 , 𝑏 = 0
  • E 𝑏 = βˆ’ 5 , π‘Ž = βˆ’ 1 2 , 𝑏 = 0

Q5:

Charlotte prepared 42 sandwiches for her guests. Write a function rule that relates the number of sandwiches per guest to the number of guests. Let 𝑔 represent the number of guests and represent 𝑐 the number of sandwiches per guest.

  • A 𝑐 = 4 2 𝑔
  • B 𝑐 = 4 2 + 𝑔
  • C 𝑐 = 4 2 βˆ’ 𝑔
  • D 𝑐 = 4 2 Γ· 𝑔
  • E 𝑐 = 𝑔 Γ· 4 2

Q6:

The given table shows the fee for borrowed books that are overdue at a library. Determine which expression could be used to find the fee for a book that is 𝑛 weeks overdue.

Weeks Overdue 1 2 3 𝑛
Fee ($) 9 11 13 ?
  • A 2 𝑛
  • B 7 𝑛 + 2
  • C 2 𝑛 + 7
  • D 7 𝑛 βˆ’ 2
  • E 2 𝑛 βˆ’ 7

Q7:

Matthew is trying to choose a car rental company. The relationship between the number of rental days and the total cost of the rental is linear for each company. Write the corresponding linear functions that describe each car rental company’s total cost.

Company A $40 per day $35 cleaning and maintenance fees
Company B 2 days cost $70 7 days cost $280
  • Acompany A: 𝑦=35π‘₯+40, company B: 𝑦=42π‘₯βˆ’14
  • Bcompany A: 𝑦=40π‘₯+35, company B: 𝑦=42π‘₯βˆ’14
  • Ccompany A: 𝑦=40π‘₯+354, company B: 𝑦=βˆ’14π‘₯+42
  • Dcompany A: 𝑦=140+35, company B: 𝑦=42π‘₯βˆ’14
  • Ecompany A: 𝑦=40π‘₯+35, company B: π‘₯42βˆ’14

Q8:

At a flea market, a woman sells her homemade hacky sacks. She paid $10.00 to have a stall erected at the market and makes $20.25 for every hacky sack she sells. Write a function for the money she earns.

  • A 𝑦 = 2 0 . 2 5 π‘₯ βˆ’ 1 0
  • B 𝑦 = 1 0 π‘₯ βˆ’ 2 0 . 2 5
  • C 𝑦 = 2 0 . 2 5 π‘₯ + 1 0
  • D 𝑦 = 3 0 . 2 5 π‘₯ βˆ’ 1 0
  • E 𝑦 = 1 0 π‘₯ + 2 0 . 2 5

Q9:

A group of friends want to go cycling and are trying to choose a bike rental company. Rental A charges $10 per hour plus a rental fee of $5, whereas rental B charges $7 for 1 hour and $25 for 4 hours. Given that the relationship between the number of hours and the cost is linear for both bike rental companies, write the corresponding linear functions that describe the total cost of each company.

  • Arental A: 𝑦=10π‘₯+5, rental B: 𝑦=6π‘₯+1
  • Brental A: 𝑦=10π‘₯βˆ’5, rental B: 𝑦=6π‘₯+1
  • Crental A: 𝑦=5π‘₯+10, rental B: 𝑦=16π‘₯+16
  • Drental A: 𝑦=10π‘₯+5, rental B: 𝑦=6π‘₯βˆ’1
  • Erental A: 𝑦=5π‘₯+10, rental B: 𝑦=π‘₯+6

Q10:

Hannah is trying to decide which pool to go swimming in. There are two nearby pools: pool A, which has a membership fee of $150 and charges $7 per visit, and pool B, which charges $260 for 15 visits and $316 for 22 visits. Write the corresponding linear functions that describe each pool’s total cost for π‘₯ swims.

  • Apool A: 𝑦=7π‘₯+150, pool B: 𝑦=π‘₯8+140
  • Bpool A: 𝑦=7π‘₯βˆ’150, pool B: 𝑦=8π‘₯βˆ’140
  • Cpool A: 𝑦=7π‘₯+150, pool B: 𝑦=8π‘₯+140
  • Dpool A: 𝑦=150π‘₯+7, pool B: 𝑦=140π‘₯+8
  • Epool A: 𝑦=7π‘₯βˆ’150, pool B: 𝑦=8π‘₯+140

Q11:

By using the linear relation π‘Ž+3𝑏=2, fill in the missing values in the table.

π‘Ž 5 β‹― βˆ’ 1 0
𝑏 β‹― 1 β‹―
  • A 𝑏 = βˆ’ 1 3 , π‘Ž = βˆ’ 1 3 , 𝑏 = 3 2
  • B 𝑏 = βˆ’ 1 , π‘Ž = 5 , 𝑏 = 4
  • C 𝑏 = βˆ’ 1 3 , π‘Ž = 1 3 , 𝑏 = 3 2
  • D 𝑏 = βˆ’ 1 , π‘Ž = βˆ’ 1 , 𝑏 = 4

Q12:

Write a function that describes the distance 𝑦Daniel jogs after xminutes.

Time (min) 5 10 20
Distance Jogged (mi) 0.75 1.5 3
  • A 𝑦 = 0 . 7 5 π‘₯
  • B 𝑦 = 0 . 1 5 π‘₯
  • C 𝑦 = βˆ’ 0 . 1 5 π‘₯
  • D 𝑦 = 1 . 5 π‘₯ + 3
  • E 𝑦 = 0 . 7 5 π‘₯ βˆ’ 3

Q13:

In this function machine, we put π‘₯ in and get 𝑦 out.

Which of these equations describes the rule?

  • A 𝑦 = π‘₯ Γ— 5
  • B 𝑦 = π‘₯ Γ— 8
  • C 𝑦 = π‘₯ + 5
  • D 𝑦 = π‘₯ βˆ’ 5
  • E 𝑦 = π‘₯ + 8

Q14:

In this function machine, we put π‘₯ in and get 𝑦 out.

It uses the rule add 20 to π‘₯ to calculate 𝑦.

If we put in 3, what number do we get out?

If we get out 32, what number did we put in?

Which of these equations describes the rule?

  • A 𝑦 = π‘₯ βˆ’ 2 0
  • B 𝑦 = π‘₯ + 2 0
  • C 𝑦 = π‘₯ + 2 3
  • D 𝑦 = π‘₯ Γ— 2 0
  • E 𝑦 = π‘₯ βˆ’ 2 3

Q15:

Write the equation that expresses 𝑦 in terms of π‘₯ for the numbers in the table.

π‘₯ 0 1 2 3
𝑦 7 8 9 10
  • A 𝑦 = π‘₯ + 7
  • B 𝑦 = 7 π‘₯
  • C 𝑦 = 3 π‘₯ βˆ’ 1 0
  • D 𝑦 = π‘₯ βˆ’ 7
  • E 𝑦 = 3 π‘₯ + 1 0

Q16:

At Elmwood Middle School, sixth graders spend 3 hours every night studying, seventh graders spend 4 hours, eighth graders spend 5 hours. Let the students’ grade be the input (π‘₯), what is the function rule between the students’ grade and the amount of time the students spend on homework every night?

  • A 𝑑 = 3 ( π‘₯ + 1 )
  • B 𝑑 = π‘₯ βˆ’ 3
  • C 𝑑 = 3 ( π‘₯ βˆ’ 1 )
  • D 𝑑 = π‘₯ + 3
  • E 𝑑 = 3 π‘₯

Q17:

A supermarket deducts $13 off the total purchase for customers visiting between 5 am and 6 am. Find a function rule that relates the final cost to the total purchase amount. Let the total purchase amount be denoted by 𝑝 and the final cost by 𝑐.

  • A 𝑐 = 𝑝 + 1 3
  • B 𝑐 = 𝑝 βˆ’ 1 3
  • C 𝑐 = 𝑝 1 3
  • D 𝑐 = 1 3 𝑝
  • E 𝑐 = 1 3 𝑝 βˆ’ 1 3

Q18:

What is the mathematical relation between 𝑦 and π‘₯ for the values in the table?

Input (π‘₯) 16 63 87
Output (𝑦) 13 60 84
  • A 𝑦 = 3 π‘₯
  • B 𝑦 = π‘₯ Γ· 3
  • C 𝑦 = π‘₯ + 3
  • D 𝑦 = π‘₯ βˆ’ 3

Q19:

The table shows the relationship between the number of gas canisters bought and the total cost.

Number Bought 10 15 20 30
Total Cost $25.00 $37.50 $50.00 $75.00

Write an equation for the total cost, 𝑇, of buying 𝑁 gas canisters.

  • A 𝑇 = 5 2 𝑁
  • B 𝑇 = 2 5 𝑁
  • C 𝑇 = 2 5 𝑁
  • D 𝑇 = 1 5 4 𝑁
  • E 𝑇 = 1 0 𝑁

Q20:

Write the equation that expresses 𝑦 in terms of π‘₯ for the numbers in the table.

π‘₯ 0 1 2 3
𝑦 6 5 4 3
  • A 𝑦 = π‘₯ βˆ’ 6
  • B 𝑦 = π‘₯ + 6
  • C 𝑦 = βˆ’ π‘₯ + 5
  • D 𝑦 = π‘₯ + 5
  • E 𝑦 = βˆ’ π‘₯ + 6

Q21:

What is the mathematical relation between 𝑦 and π‘₯ for the values in the table?

Input (π‘₯) 117 450 144
Output (𝑦) 13 50 16
  • A 𝑦 = π‘₯ Γ· 9
  • B 𝑦 = 9 π‘₯
  • C 𝑦 = π‘₯ βˆ’ 9
  • D 𝑦 = π‘₯ + 9

Q22:

The table represents a proportional relationship between π‘₯ and 𝑦. Write an equation to describe the relationship between π‘₯ and 𝑦.

π‘₯ 1 2 1 3 2 2
𝑦 1 2 3 4
  • A 𝑦 = 3 π‘₯
  • B π‘₯ = 3 𝑦
  • C π‘₯ = 2 𝑦
  • D 𝑦 = 2 π‘₯
  • E 𝑦 = π‘₯ + 2

Q23:

Write the equation that expresses 𝑦 in terms of π‘₯ for the numbers in the table.

π‘₯ 2 3 4 5
𝑦 1 6 11 16
  • A 𝑦 = 3 π‘₯ + 1
  • B 𝑦 = π‘₯ βˆ’ 1
  • C 𝑦 = 5 π‘₯ βˆ’ 9
  • D 𝑦 = π‘₯ + 1
  • E 𝑦 = 2 π‘₯ + 1

Q24:

Use the table to write an expression for the total cost, 𝑇, of buying 𝑁 gas canisters.

Number Bought 10 12 15 20
Total Cost $ 2 5 . 0 0 $ 3 0 . 0 0 $ 3 7 . 5 0 $ 5 0 . 0 0
  • A 𝑇 = 3 . 7 5 𝑁
  • B 𝑇 = 1 0 . 0 0 𝑁
  • C 𝑇 = 2 . 5 0 𝑁
  • D 𝑇 = 2 5 . 0 0 𝑁
  • E 𝑇 = 0 . 4 0 𝑁

Q25:

James sold glasses of lemonade during the spring and summer to raise money to buy a car. The table shows how many glasses he sold each month as well as the profit he made. Write an equation which shows the relationship between the number of glasses sold (𝑛) and the profit made (𝑝).

March April May June July August September
Glasses Sold (𝑛) 8 3 2 12 10 5 4
Profit (𝑝) $36.00 $13.50 $9.00 $54.00 $45.00 $22.00 $18.00
  • A 𝑛 = 1 3 . 5 𝑝
  • B 𝑝 = 1 3 . 5 𝑛
  • C 𝑝 = 4 . 5 𝑛
  • D 𝑝 = 9 𝑛
  • E 𝑛 = 4 . 5 𝑝

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