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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. Set the denominator equal to 0.
  4. x - 3 = 0 gives x = 3.
  5. That input is excluded.
  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. Set the denominator equal to 0.
  4. x - 3 = 0 gives x = 3.
  5. That input is excluded.
  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. That input is excluded.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

3. What is the domain condition for f(x) = sqrt(x - 5)?

Choices: x >= 5 · x > 5 · x <= 5 · x != 5

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

Answer: x >= 5

4. What is the domain condition for 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 condition for 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 the radicand to be nonnegative.
  4. 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 condition for f(x) = log(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 condition for f(x) = log(x + 7)?
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. A logarithm input must be positive.
  4. 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

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