CMClearMathAcademy

Domain and Range from Equations

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

Domain comes from what inputs are allowed before any graphing happens. 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. What value is excluded from the domain of f(x) = 1/(x - 3)?

  1. Worked Example: First identify exactly what the question is asking: What value is excluded from the domain of f(x) = 1/(x - 3)?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. A fraction is undefined where its denominator is 0.
  4. x - 3 = 0 gives x = 3.
  5. So 3 is excluded from the domain.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

Practice problems

1. What value is excluded from the domain of f(x) = 1/(x - 3)?

Show solution
  1. Warm-up: First identify exactly what the question is asking: What value is excluded from the domain of f(x) = 1/(x - 3)?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. A fraction is undefined where its denominator is 0.
  4. x - 3 = 0 gives x = 3.
  5. So 3 is excluded from the domain.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

2. What value is excluded from the domain of f(x) = 1/(x + 4)?

Show solution
  1. Warm-up: First identify exactly what the question is asking: What value is excluded from the domain of f(x) = 1/(x + 4)?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Set the denominator equal to 0.
  4. x + 4 = 0 gives x = -4.
  5. So -4 is excluded from the domain.
  6. Check the result by substituting or estimating: the response should match -4 and make sense in the original problem.

Answer: -4

3. What value is excluded from the domain of f(x) = 1/(4x - 20)?

Show solution
  1. Core Practice: First identify exactly what the question is asking: What value is excluded from the domain of f(x) = 1/(4x - 20)?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Set the denominator equal to 0.
  4. 4x - 20 = 0 gives 4x = 20, so x = 5.
  5. So 5 is excluded.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

4. What is the domain of f(x) = sqrt(x - 6)?

Choices: x >= 6 · x > 6 · x <= 6 · x != 6

Show solution
  1. Core Practice: First identify exactly what the question is asking: What is the domain of f(x) = sqrt(x - 6)?
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. A square root needs a nonnegative radicand.
  4. Require x - 6 >= 0.
  5. So x >= 6.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x >= 6

5. What is the domain of f(x) = sqrt(x + 7)?

Choices: x >= -7 · x > -7 · x <= -7 · x != -7

Show solution
  1. Core Practice: First identify exactly what the question is asking: What is the domain of f(x) = sqrt(x + 7)?
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. A square root needs a nonnegative radicand.
  4. Require x + 7 >= 0.
  5. So x >= -7.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x >= -7

Practice this interactively with instant feedback and an AI tutor.

Practice Domain and Range from Equations Take the free placement check

More Precalculus lessons