Math worksheet designed to help students practice completing function tables for linear equations.

Problem Analysis:

The image contains a worksheet titled "Function Table - Linear Function." The task involves completing function tables and solving problems related to linear functions. Let's break it down step by step.

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Part A: Complete each function rule

The table provides some input-output pairs, and we need to determine the function rule for each pair.

#### 1. First Row:
- Inputs: $ x = 0, 1, 2, 3 $
- Outputs: $ f(x) = 4, 7, 10, 13 $

To find the function rule:
- Calculate the difference between consecutive outputs:
$$
7 - 4 = 3, \quad 10 - 7 = 3, \quad 13 - 10 = 3
$$
The outputs increase by 3 for each increment of 1 in $ x $. This suggests a linear function of the form:
$$
f(x) = mx + b
$$
where $ m = 3 $ (the slope).

- To find the y-intercept $ b $, use the point $ (0, 4) $:
$$
f(0) = 3(0) + b = 4 \implies b = 4
$$

Thus, the function rule is:
$$
f(x) = 3x + 4
$$

#### 2. Second Row:
- Inputs: $ x = 0, 1, 2, 3 $
- Outputs: $ f(x) = 5, 8, 11, 14 $

To find the function rule:
- Calculate the difference between consecutive outputs:
$$
8 - 5 = 3, \quad 11 - 8 = 3, \quad 14 - 11 = 3
$$
The outputs increase by 3 for each increment of 1 in $ x $. This suggests a linear function of the form:
$$
f(x) = mx + b
$$
where $ m = 3 $ (the slope).

- To find the y-intercept $ b $, use the point $ (0, 5) $:
$$
f(0) = 3(0) + b = 5 \implies b = 5
$$

Thus, the function rule is:
$$
f(x) = 3x + 5
$$

#### 3. Third Row:
- Inputs: $ x = 0, 1, 2, 3 $
- Outputs: $ f(x) = 6, 9, 12, 15 $

To find the function rule:
- Calculate the difference between consecutive outputs:
$$
9 - 6 = 3, \quad 12 - 9 = 3, \quad 15 - 12 = 3
$$
The outputs increase by 3 for each increment of 1 in $ x $. This suggests a linear function of the form:
$$
f(x) = mx + b
$$
where $ m = 3 $ (the slope).

- To find the y-intercept $ b $, use the point $ (0, 6) $:
$$
f(0) = 3(0) + b = 6 \implies b = 6
$$

Thus, the function rule is:
$$
f(x) = 3x + 6
$$

#### Final Answer for Part A:
$$
\boxed{f(x) = 3x + 4, \quad f(x) = 3x + 5, \quad f(x) = 3x + 6}
$$

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Part B: Complete the function table using the rule $ f(x) = 3x + 4 $

We are given the function rule $ f(x) = 3x + 4 $ and need to complete the table for specific values of $ x $.

#### 1. When $ x = -2 $:
$$
f(-2) = 3(-2) + 4 = -6 + 4 = -2
$$

#### 2. When $ x = 0.5 $:
$$
f(0.5) = 3(0.5) + 4 = 1.5 + 4 = 5.5
$$

#### 3. When $ x = 1007 $:
$$
f(1007) = 3(1007) + 4 = 3021 + 4 = 3025
$$

#### Final Answers for Part B:
$$
\boxed{-2, 5.5, 3025}
$$

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Part C: Complete the function table

We are given the function rule $ f(x) = -2x + 3 $ and need to complete the table for specific values of $ x $.

#### 1. When $ x = -2 $:
$$
f(-2) = -2(-2) + 3 = 4 + 3 = 7
$$

#### 2. When $ x = 0.5 $:
$$
f(0.5) = -2(0.5) + 3 = -1 + 3 = 2
$$

#### Final Answers for Part C:
$$
\boxed{7, 2}
$$

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Summary of Final Answers:

- Part A: $ f(x) = 3x + 4, \quad f(x) = 3x + 5, \quad f(x) = 3x + 6 $
- Part B: $ -2, 5.5, 3025 $
- Part C: $ 7, 2 $

$$
\boxed{f(x) = 3x + 4, \quad f(x) = 3x + 5, \quad f(x) = 3x + 6, \quad -2, 5.5, 3025, \quad 7, 2}
$$