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Domain and Range

A free Algebra I lesson from the “Functions, Linear Relationships, and Rate of Change” unit, with a worked example and practice problems including step-by-step solutions.

The domain is the set of allowed inputs; the range is the set of outputs actually produced from those inputs. Start with the rule and its number system: in real-number algebra, a denominator cannot be zero and a square-root radicand must be nonnegative. These restrictions are combined, not considered separately. Then apply the context: a ticket count is a whole number, while a time measurement can vary continuously over an interval. A formula such as C(n) = 8n may be algebraically defined for all real numbers, but a purchase of at most 12 tickets uses only n = 0, 1, ..., 12. Domain restrictions can also change the range. For f(x) = (x - 2)^2 on 0 <= x <= 5, the minimum is 0 at x = 2 and the maximum is 9 at x = 5. Do not merely square the domain's endpoints and assume the smaller result is the minimum: a turning point may lie between them. Include an endpoint when its input is permitted and exclude it when it is not.

What you'll learn

Why it matters: A capacity limit determines which purchases are possible. The same equation may model an uninterrupted measurement or only whole-number choices, producing different graphs and ranges.

Worked example

Problem. A seller has 12 tickets, priced at $8 each. Let R(n) = 8n be the revenue after selling n tickets. Give the domain and range, and explain whether (2.5, 20) belongs on this model's graph.

  1. Define n as the number of tickets sold; it must be a whole number.
  2. The inventory limits n to 0 through 12, including both endpoints.
  3. Apply R(n) = 8n to those inputs: the possible revenues are 0, 8, 16, and so on through 96 dollars.
  4. Although 8(2.5) = 20, the input 2.5 is not allowed because this seller does not sell fractional tickets.
  5. The model's graph consists of separate points, not the entire line segment.

Answer: Domain: {0, 1, ..., 12}; range: {0, 8, 16, ..., 96}; (2.5, 20) is not part of the model.

Lesson preview: this page shows 5 practice items. The interactive activity contains 32 practice items and 17 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.

Practice problems

1. What is the real-number domain of f(x) = sqrt(x + 2)/(x - 1)?

Choices: x >= -2 and x != 1 · x >= -2 · x > 1 · All real x except -2

Show solution
  1. The square root requires x + 2 >= 0, or x >= -2.
  2. The denominator also requires x - 1 != 0, so x != 1.
  3. Both conditions must hold at the same time.

Answer: x >= -2 and x != 1

2. For f(x) = (x - 2)^2 with domain 0 <= x <= 5, what is the minimum output?

Show solution
  1. A square is nonnegative.
  2. The value 0 occurs at x = 2, and 2 is in the allowed domain.
  3. The minimum is therefore 0, even though the endpoint outputs are 4 and 9.

Answer: 0

3. A learner models the number of cars in a 20-space lot using every real input from 0 to 20. What restriction is missing?

Choices: The inputs must be whole numbers. · The inputs must be strictly greater than 20. · The inputs must exclude 0. · The inputs must be negative.

Show solution
  1. The lot may contain 0 through 20 cars.
  2. A count of cars does not include fractional values such as 3.7.

Answer: The inputs must be whole numbers.

4. The rule h(t) = 9 - (t - 3)^2 models height for 0 <= t <= 6. What is the largest height in this model?

Show solution
  1. A square is nonnegative, so subtracting (t - 3)^2 cannot give an output above 9.
  2. The largest output is 9 when the squared term is 0, at t = 3.
  3. That input is inside the allowed interval.

Answer: 9

5. For f(x) = 1/(x - 4), enter the excluded x-value.

Show solution
  1. x - 4 = 0 means x = 4 is excluded.

Answer: 4

Explain and apply

Complete these tasks on paper. Use the rubric to review your reasoning with a teacher or study partner; these tasks are not automatically scored.

Justify the whole range

A model uses f(x) = (x - 3)^2 + 1 for 1 <= x <= 4. Determine its domain and range, identify where each extreme occurs, and explain why checking only the two endpoints would give the wrong minimum.

Review rubric and solution
  • State the given domain [1, 4].
  • Use the vertex at x = 3 to obtain the minimum output 1.
  • Compare endpoint outputs 5 and 2 to find the maximum 5.
  • Conclude range [1, 5] and explain why the vertex matters.
  1. The domain is [1, 4].
  2. Since (x - 3)^2 is nonnegative and x = 3 is allowed, the minimum is f(3) = 1.
  3. The endpoints give f(1) = 5 and f(4) = 2. The input farthest from 3 is 1, so the maximum is 5.
  4. The continuous quadratic takes every output between these extremes, giving range [1, 5]. Checking only the endpoints would incorrectly suggest a minimum of 2.

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