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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's points run from x = -3 to x = 4 (both included). What is the domain?

  1. Worked Example: First identify exactly what the question is asking: A graph's points run from x = -3 to x = 4 (both included). 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 interval of x-values.
  4. The graph spans x = -3 to x = 4.
  5. Filled endpoints use square brackets.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: [-3, 4]

Practice problems

1. A graph's points run from x = -3 to x = 4 (both included). What is the domain?

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: A graph's points run from x = -3 to x = 4 (both included). 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 interval of x-values.
  4. The graph spans x = -3 to x = 4.
  5. Filled endpoints use square brackets.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: [-3, 4]

2. A graph's y-values run from -3 to 6 (both included). What is the range?

Choices: [-3, 6] · [-4, 6] · [-3, 7] · [-2, 5]

Show solution
  1. Warm-up: First identify exactly what the question is asking: A graph's y-values run from -3 to 6 (both included). 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 interval of y-values.
  4. The lowest y is -3 and the highest is 6.
  5. Read the range from 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: [-3, 6]

3. A hollow (open) endpoint on a graph means that x-value is:

Choices: not included in the domain · included in the domain · the maximum output · the y-intercept

Show solution
  1. An open circle marks a value the graph approaches but does not reach.
  2. It is excluded from the domain.
  3. A filled circle would include it.

Answer: not included in the domain

4. A filled (closed) endpoint on a graph means that x-value is:

Choices: included in the domain · excluded from the domain · an asymptote · always zero

Show solution
  1. Core Practice: First identify exactly what the question is asking: A filled (closed) endpoint on a graph means that x-value is:
  2. For data questions, identify what each statistic measures before calculating so the result matches the question.
  3. A solid dot marks a value the graph actually reaches.
  4. It is part of the domain.
  5. An open circle would exclude it.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: included in the domain

5. A graph's points run from x = -3 to x = 3. What is the smallest x-value in the domain?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A graph's points run from x = -3 to x = 3. What is the smallest x-value in the domain?
  2. For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
  3. The domain is the set of x-values covered.
  4. It starts at the left edge, x = -3.
  5. So the smallest x-value is -3.
  6. Check the result by substituting or estimating: the response should match -3 and make sense in the original problem.

Answer: -3

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