Note: If the mapping has equal interval (difference) for the y and x interval of 1, you can use the general linear sequence formula to find the rule of the mapping.

U_n = U₁ + (n-1)d

In the mapping, U_n = y and n = x and U₁, the first value of y

U₁ = 4

y = 4 + (x-1)d

The d is the interval between each consecutive y

In this mapping, d = 2-4 = 0-2 = -2 - 0 = -4 -(-2) = -2

We can substitute our d and simplify the expression for y

y = 4 + (x-1)d

y = 4 + (x-1)(-2)

y = 4 + -2 x x -1 x -2

y = 4 + -2x + 2

y = 4 + 2 -2x

y = 6 + -2x

y = -2x + 6

Substitute x by 2 in the equation and solve for the expression

x → 2x² - 1

2(2)² - 1 = 2 x 4 - 1 = 8 - 1 = 7

NOTE: x² = x x x

(2)² = 2² = 2 x 2 = 4

Check if the mapping is a linear/arithmetic sequence by finding the difference between each consecutive value of y. A linear sequence has the common difference between consecutive numbers the same

If it is linear, use the equation of linear formula to find the rule of the mapping

y₂ - y₁ = 9 - 5 = 4

y₃ - y₂ = 13 - 9 = 4

y₄ - y₃ = 17- 13 = 4

y₅ - y₄ = 21 - 17 = 4

As you can see, the rule of the mapping is a linear since the difference is the same, hence use the linear sequence formula to find the rule

U_n = U₁ + (n-1)d

Where U_n is the value at the n^th position, in the rule of mapping, the y value

U₁ is the first term, in the mapping above, 5

d is the common difference between each consecutive terms

n is the position, in the case of the above mapping, x

y = 5 + (x-1)4

y = 5 + (x-1) x 4

y = 5 + x x 4 -1x4

y = 5 + 4x-4

y = 4x+5-4

y = 4x+1

We can test with when x = 3

y = 4x+1

y = 4 x 3+1

y = 12+1 = 13

You can try the values of x to see if you will get the corresponding y value to prove the rule is correct

Find the rule of the mapping first

Check if the mapping is a linear/arithmetic sequence by finding the difference between each consecutive value of y. A linear sequence has the common difference between consecutive numbers the same

If it is linear, use the equation of linear formula to find the rule of the mapping

y₂ - y₁ = 9 - 5 = 4

y₃ - y₂ = 13 - 9 = 4

y₄ - y₃ = 17- 13 = 4

y₅ - y₄ = 21 - 17 = 4

As you can see, the rule of the mapping is a linear since the difference is the same, hence use the linear sequence formula to find the rule

U_n = U₁ + (n-1)d

Where U_n is the value at the n^th position, in the rule of mapping, the y value

U₁ is the first term, in the mapping above, 5

d is the common difference between each consecutive terms

n is the position, in the case of the above mapping, x

y = 5 + (x-1)4

y = 5 + (x-1) x 4

y = 5 + x x 4 -1x4

y = 5 + 4x-4

y = 4x+5-4

y = 4x+1

We can test with when x = 3

y = 4x+1

y = 4 x 3+1

y = 12+1 = 13

You can try the values of x to see if you will get the corresponding y value to prove the rule is correct

y = 4x+1

y = 37

37 = 4x+1

Make x the subject

37 - 1 = 4x

36 = 4x

Divide both sides by 4

36/4 = 4x/4

9 = x

Substitute the value of x in each of the rule to see which one gives you the corresponding y value in all the cases

Only x→ x²+1 gives the corresponding y values

Thus when x = 1 → 1²+1 = 1+1 = 2

When x = 2 → 2²+1 = 4+1 = 5

When x = 3 → 3²+1 = 9+1 = 10

When x = 4 → 4²+1 = 16+1 = 17

Find the rule of the mapping first

Check if the mapping is a linear/arithmetic sequence by finding the difference between each consecutive value of y. A linear sequence has the common difference between consecutive numbers the same

If it is linear, use the equation of linear formula to find the rule of the mapping

y₂ - y₁ = 9 - 5 = 4

y₃ - y₂ = 13 - 9 = 4

y₄ - y₃ = 17- 13 = 4

y₅ - y₄ = 21 - 17 = 4

As you can see, the rule of the mapping is a linear since the difference is the same, hence use the linear sequence formula to find the rule

U_n = U₁ + (n-1)d

Where U_n is the value at the n^th position, in the rule of mapping, the y value

U₁ is the first term, in the mapping above, 5

d is the common difference between each consecutive terms

n is the position, in the case of the above mapping, x

y = 5 + (x-1)4

y = 5 + (x-1) x 4

y = 5 + x x 4 -1x4

y = 5 + 4x-4

y = 4x+5-4

y = 4x+1

We can test with when x = 3

y = 4x+1

y = 4 x 3+1

y = 12+1 = 13

You can try the values of x to see if you will get the corresponding y value to prove the rule is correct

y = 4x+1

y = 37

37 = 4x+1

Make x the subject

37 - 1 = 4x

36 = 4x

Divide both sides by 4

36/4 = 4x/4

9 = x

Check if the mapping is a linear/arithmetic sequence by finding the difference between each consecutive value of y. A linear sequence has the common difference between consecutive numbers the same

If it is linear, use the equation of linear formula to find the rule of the mapping

y₂ - y₁ = 9 - 5 = 4

y₃ - y₂ = 13 - 9 = 4

y₄ - y₃ = 17- 13 = 4

y₅ - y₄ = 21 - 17 = 4

As you can see, the rule of the mapping is a linear since the difference is the same, hence use the linear sequence formula to find the rule

U_n = U₁ + (n-1)d

Where U_n is the value at the n^th position, in the rule of mapping, the y value

U₁ is the first term, in the mapping above, 5

d is the common difference between each consecutive terms

n is the position, in the case of the above mapping, x

y = 5 + (x-1)4

y = 5 + (x-1) x 4

y = 5 + x x 4 -1x4

y = 5 + 4x-4

y = 4x+5-4

y = 4x+1

We can test with when x = 3

y = 4x+1

y = 4 x 3+1

y = 12+1 = 13

You can try the values of x to see if you will get the corresponding y value to prove the rule is correct

Note: If the mapping has equal interval (difference) for the y and x interval of 1, you can use the general linear sequence formula to find the rule of the mapping.

U_n = U₁ + (n-1)d

In the mapping, U_n = y and n = x and U₁ is the value of y when x = 1

U₁ = 3

y = 3 + (x-1)d

The d is the interval between each consecutive y

In this mapping, d = 6-3 = 9-6 = 12-9 = 15-12 = 3

We can substitute our d and simplify the expression for y

y = 3 + (x-1)(3)

y = 3 + 3(x) -1(3)

y = 3 + 3x - 3

y = 3x

y → 3x

Notes:

1. If you decide to substitute values for x for each of the options provided, check for all values of x.
2. Just checking once won't be sufficient.
3. For example, in x → 4 - x, substituting x = 1 gives 4 - 1 = 3, which matches the first value in the mapping but not the others hence it is not the correct rule.
4. Likewise, in x → x + 2, substituting x = 1 gives 1 + 2 = 3, which matches the first value in the mapping but not the others hence it is not the correct rule.