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Domain and Range from Graphs

A free Precalculus lesson from the “Function Foundations and Behavior” unit, with a worked example and practice problems including step-by-step solutions.

Domain reads left to right across x-values; range reads bottom to top across y-values. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Function behavior is the language behind Calculus, physics graphs, economic models, and computer-generated curves.

Worked example

Problem. A graph has x-values from -3 to 5. What is the domain?

  1. Worked Example: First identify exactly what the question is asking: A graph has x-values from -3 to 5. What is the domain?
  2. For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
  3. Domain is the x-value interval.
  4. The graph starts at x = -3 and ends at x = 5.
  5. Use brackets when endpoints are included.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: [-3, 5]

Practice problems

1. A graph has x-values from -3 to 5. What is the domain?

Choices: [-3, 5] · [-2, 4] · [-3, 4] · [-2, 5]

Show solution
  1. Warm-up: First identify exactly what the question is asking: A graph has x-values from -3 to 5. What is the domain?
  2. For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
  3. Domain is the x-value interval.
  4. The graph starts at x = -3 and ends at x = 5.
  5. Use brackets when endpoints are included.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: [-3, 5]

2. A graph has x-values from -4 to 6. What is the domain?

Choices: [-4, 6] · [-3, 5] · [-4, 5] · [-3, 6]

Show solution
  1. Warm-up: First identify exactly what the question is asking: A graph has x-values from -4 to 6. What is the domain?
  2. For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
  3. Domain is the x-value interval.
  4. The graph starts at x = -4 and ends at x = 6.
  5. Use brackets when endpoints are included.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: [-4, 6]

3. A graph has y-values from -1 to 6. What is the range?

Choices: [-1, 6] · [-5, 7] · [-5, 6] · [-1, 7]

Show solution
  1. Core Practice: First identify exactly what the question is asking: A graph has y-values from -1 to 6. What is the range?
  2. For range questions, identify the possible output values after the input restrictions and graph shape are considered.
  3. Range is the y-value interval.
  4. The lowest y is -1 and the highest y is 6.
  5. Read range bottom to top.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: [-1, 6]

4. A graph has y-values from -2 to 3. What is the range?

Choices: [-2, 3] · [-2, 8]

Show solution
  1. Core Practice: First identify exactly what the question is asking: A graph has y-values from -2 to 3. What is the range?
  2. For range questions, identify the possible output values after the input restrictions and graph shape are considered.
  3. Range is the y-value interval.
  4. The lowest y is -2 and the highest y is 3.
  5. Read range bottom to top.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: [-2, 3]

5. A hollow endpoint on a graph means the endpoint is:

Choices: not included · included · the maximum only · the y-intercept

Show solution
  1. A hollow point means the graph approaches that value but does not include it.
  2. Use a parenthesis for a hollow endpoint.
  3. A filled point would be included.

Answer: not included

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