Solved Examples and Worksheet for Finding the Rule of a Function Table

Q1Which of the following rules describes the best input - output relationship from the table?
InputOutput
65
1110
1514
76

A. Output = 3 - Input
B. Output = 1 - Input
C. Output = Input - 1
D. Output = Input + 1

Solutions
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Step: 1
On observing the table, it is found that the output 5 = 6 - 1, 10 = 11 - 1, 14 = 15 - 1 and 6 = 7 - 1
Step: 2
So, the relation between the input and output values is Output = Input - 1.
Correct Answer is :   Output = Input - 1
Q2Mr. Nathan requires x gallons of milk for their family. The cost of x gallons of milk is given by f(x) = 1.19x. Which of the tables best suits the equation?

A. Table 1
B. Table 2
C. Table 3
D. Table 4

Solutions
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Step: 1
f(x) = 1.19x
  [Original equation.]
Step: 2
f(1)= 1.19(1) = 1.19
  [Substitute 1 for x and simplify.]
Step: 3
f(2) = 1.19(2) = 2.38
  [Substitute 2 for x and simplify.]
Step: 4
f(3) = 1.19(3) = 3.57
  [Substitute 3 for x and simplify.]
Step: 5
f(4)= 1.19(4) = 4.76
  [Substitute 4 for x and simplify.]
Step: 6
Table 3 matches with the equation.
Correct Answer is :   Table 3
Q3Write a rule for the linear function in the table.


A. f(x) = 4x - 1
B. f(x) = 3x + 1
C. f(x) = 2x + 3
D. None of the above

Solutions
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Step: 1
The formula to write a function rule from a table is f(x) = mx + b.
Step: 2

Step: 3
The value of x increases by 3 and the f(x) value increases by 12.
Step: 4
m = 123 = 4
  [m = difference between f(x) values / difference between x values.]
Step: 5
When x = 0, f(x) = -1
Step: 6
b = -1
  [Since b is the value of f(x) when x = 0.]
Step: 7
The rule for the linear function is f(x) = 4x - 1.
Correct Answer is :   f(x) = 4x - 1
Q4Which of the following rules describes the input - output relationship from the table?
InputOutput
55
66
77
88
99

A. Output = Input2
B. Output = Input + 1
C. Output = Input - 1
D. Output = Input

Solutions
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Step: 1
On observing the table it is found that the output 5 = 5, 6 = 6, 7 = 7, 8 = 8 and 9 = 9.
Step: 2
So, we can write the relationship between input and output as a rule, Output = input.
Correct Answer is :   Output = Input
Q5Which equation satisfies the input-output table?
Input(x)012345
Output(y)504846444240

A. y = 50 - 3x
B. y = 50 - 2x
C. y = 50 - 4x
D. y = 50 - 5x

Solutions
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Step: 1
y = 50 - 3x
  [Consider choice A.]
Step: 2
y = 50 - 3(0) = 50
  [Replace x with 0.]
Step: 3
y = 50 - 3(1) = 47
  [Replace x with 1.]
Step: 4
In the table for x = 1, y = 48. So first equation is ruled out.
Step: 5
y = 50 - 2x
  [Consider choice B.]
Step: 6
y = 50 - 2(0) = 50
  [Replace x with 0.]
Step: 7
y = 50 - 2(1) = 48
  [Replace x with 1.]
Step: 8
y = 50 - 2(2) = 46
  [Replace x with 2.]
Step: 9
y = 50 - 2(3) = 44
  [Replace x with 3.]
Step: 10
y = 50 - 2(4) = 42
  [Replace x with 4.]
Step: 11
y = 50 - 2(5) = 40
  [Replace x with 5.]
Step: 12
The equation y = 50 - 2x satisfies the table.
Correct Answer is :   y = 50 - 2x
Q6Choose a rule for the linear function in the table shown.

A. f(x) = - 2x - 1
B. f(x) = 2x + 3
C. f(x) = 12x - 1
D. f(x) = 2x + 1

Solutions
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Step: 1
The formula to write a function rule from a table is f(x) = mx + b.
Step: 2

Step: 3
The value of x increases by 1 and the f(x) value increases by 2.
Step: 4
m = 21 = 2
  [m = difference between f(x) valuesdifference between x values.]
Step: 5
When x = 0, f(x) = 1
Step: 6
b = 1
  [Since b is the value of f(x) when x = 0.]
Step: 7
The rule for the linear function is f(x) = 2x + 1.
Correct Answer is :   f(x) = 2x + 1
Q7Identify the equation that satisfies the input-output table given below.
Input(x)012345
Output(y)504642383430


A. y = 50 - 5x
B. y = 50 - 4x
C. y = 50 - 6x
D. y = 50 - 7x
E. y = 50 - 8x

Solutions
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Step: 1
Consider choice A,
y = 50 - 5x.
Step: 2
Substitute 0 for x,
50 - 5(0) = 50.
Step: 3
Substitute 1 for x,
y = 50 - 5(1) = 45.
Step: 4
In the given table for x = 1, y = 46. So first equation is ruled out.
Step: 5
Consider choice B,
y = 50 - 4x.
Step: 6
Substitute 0 for x,
y = 50 - 4(0) = 50.
Step: 7
Substitute1 for x,
y = 50 - 4(1) = 46.
Step: 8
Substitute 2 for x,
y = 50 - 4(2) = 42.
Step: 9
Substitute 3 for x,
y = 50 - 4(3) = 38.
Step: 10
Substitute 4 for x,
y = 50 - 4(4) = 34.
Step: 11
Substitute 5 for x,
y = 50 - 4(5) = 30.
Step: 12
The equation y = 50 - 4x satisfies the given table.
Correct Answer is :   y = 50 - 4x
Q8What is the rule for the following table?
AB
542530
426--
368356
225213


A. subtract 13 from the number in column A
B. subtract 11 from the number in column A
C. subtract 12 from the number in column A
D. subtract 10 from the number in column A

Solutions
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Step: 1
The rule for the pattern is subtract 12 from the number in column A to get the number in column B.
542 - 12 = 530, 368 - 12 = 356, 225 - 12 = 213.
Step: 2
The missing number is 414.
Step: 3
So, the rule for the pattern is subtract 12 from the number in column A to get the number in column B
Correct Answer is :   subtract 12 from the number in column A
Q9Find a rule for the table.

A. multiply by 8
B. divide by 4
C. subtract 5
D. add 10

Solutions
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Step: 1
By dividing the values in column X by 4, we get the corresponding values in Column Y.
Correct Answer is :   divide by 4
Q10Identify the equation that represents the values in the function table shown.

x10253545
y2579


A. y = 5x
B. y = 5x
C. y = x5
D. y = x - 5

Solutions
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Step: 1
On observing the table it is found that 2 = 105, 5 = 255, 7 = 355, 9 = 455.
Step: 2
So, the equation that represents the table is y = x5.
Correct Answer is :   y = x5
Q11Identify the equation that represents the table.
x691215
y4567

A. y = x + 2
B. y = 3x - 1
C. y = x3 + 2
D. y = 3(x + 1)

Solutions
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Step: 1
On observing the table it is found that 4 = 63 + 2, 5 = 93 + 2, 6 = 123 + 2, 7 = 153 + 2.
Step: 2
So, the equation that represents the table is y = x3 + 2.
Correct Answer is :   y = x3 + 2
Q12What is the rule for the table?
Input3568
Output9131519


A. Add 3 and subtract 3.
B. Divide by 3 and add 2.
C. Multiply by 1 and add 2.
D. Multiply by 2 and add 3.

Solutions
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Step: 1
Consider the first input value given in the table and try each rule on the value to check if we get the corresponding output value.
Step: 2
Consider 3.
(3 + 3) - 3 = 3, not 9. So, choice 1 does not work.
Step: 3
(3 ÷ 3) + 2 = 3, not 9. So, choice 2 does not work.
Step: 4
(3 × 1) + 2 = 5, not 9. So, choice 3 does not work.
Step: 5
(3 × 2) + 3 = 9, choice 4 works.
Step: 6
So, 'multiply by 2 and add 3' is the rule for the table.
Correct Answer is :   Multiply by 2 and add 3.
Q13Andrew deposited $20 initially in his bank account. Then he deposited $15 every week. The total savings, 'c' is represented by the equation, c = 20 + 15n, where 'n' denotes the number of weeks. Which table of values satisfies the equation?

A. Table 1
B. Table 3
C. Table 4
D. Table 2

Solutions
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Step: 1
Substitute from 0 to 5 for 'n' in the equation.
Step: 2
You get the following table of values.
Step: 3
Compare the labels and values in this table with the tables given in the question.
Step: 4
Notice that only table 1 matches our table.
  [Table 4 has n = 6 instead of 5. For n = 6, c = 110.]
Correct Answer is :   Table 1
Q14The table represents the relationship between x and y.
x- 1123
y5789

Which of these equations best represents the relationship?

A. y = 3x + 3
B. y = x + 6
C. y = 2x + 2
D. y = - x - 5

Solutions
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Step: 1
The values of y obtained when x = -1, 1, 2, 3 in y = 2x + 2 are 0, 4, 6, 8.
Step: 2
The values of y obtained when x = -1, 1, 2, 3 in y = 3x + 3 are 0, 6, 9, 12.
Step: 3
The values of y obtained when x = -1, 1, 2, 3 in y = -x - 5 are -4, -6, -7, -8
Step: 4
The values of y obtained when x = -1, 1, 2, 3 in y = x + 6 are 5, 7, 8, 9.
Step: 5
The values of y obtained for the equation y = x + 6 are same as that of the values given in the table.
Step: 6
So, the equation y = x + 6 best represents the given table.
Correct Answer is :   y = x + 6
Q15Which of the equations best represents the rule for the table?
x1234
y8162432

A. y = 8x
B. y = 8x + 5
C. y = 3x + 7
D. y = x + 7

Solutions
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Step: 1
The values of y obtained when x = 1, 2, 3, 4 in y = 3x + 7 are 10, 13, 16, 19.
Step: 2
The values of y obtained when x = 1, 2, 3, 4 in y = x + 7 are 8, 9, 10, 11.
Step: 3
The values of y obtained when x = 1, 2, 3, 4 in y = 8x + 5 are 13, 21, 29, 37.
Step: 4
The values of y obtained when x = 1, 2, 3, 4 in y = 8x are 8, 16, 24, 32.
Step: 5
The values of y obtained for the equation y = 8x are same as that of the values given in the table.
Step: 6
So, the equation y = 8x best represents the given table.
Correct Answer is :   y = 8x

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