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Linear Functions Introduction

A free Pre-Algebra lesson from the “Coordinate Plane, Functions, and Slope” unit, with a worked example and practice problems including step-by-step solutions.

A function pairs each input with exactly one output. A linear function has a constant rate of change — its graph is a straight line and its equation looks like y = mx + b (or f(x) = mx + b). Function notation f(x) lets you name the rule and ask for its value at any input.

What you'll learn

Why it matters: Cell-phone plans with a flat fee plus a per-minute charge, taxi fares (drop fee plus per-mile rate), and any constant-rate scenario produce linear functions.

Worked example

Problem. If f(x) = 3x + 2, find f(4).

  1. Substitute 4 for x: f(4) = 3(4) + 2.
  2. = 12 + 2 = 14.

Answer: 14

Practice problems

1. Is y = 2x + 5 a linear function?

Choices: Yes · No

Show solution
  1. Warm-up: First identify exactly what the question is asking: Is y = 2x + 5 a linear function?
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. Form y = mx + b with m=2 and b=5.
  4. That's a linear function.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Yes

2. Is y = x^2 a linear function?

Choices: Yes · No

Show solution
  1. Warm-up: First identify exactly what the question is asking: Is y = x^2 a linear function?
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. x^2 is degree 2 (a parabola).
  4. Linear functions are degree 1.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: No

3. If f(x) = x + 7, find f(3).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = x + 7, find f(3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(3) = 3 + 7 = 10.
  4. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.

Answer: 10

4. If f(x) = 2x, find f(-3).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = 2x, find f(-3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(-3) = 2 * (-3) = -6.
  4. Check the result by substituting or estimating: the response should match -6 and make sense in the original problem.

Answer: -6

5. A table has (1,3), (2,5), (3,7), (4,9). Linear?

Choices: Yes · No

Show solution
  1. Core Practice: First identify exactly what the question is asking: A table has (1,3), (2,5), (3,7), (4,9). Linear?
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. Outputs increase by 2 each step.
  4. Constant difference means linear.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Yes

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