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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 every legal input; the range is every possible output. For a fraction expression, the denominator cannot be zero. For a square root, the input cannot be negative. From a set of ordered pairs, the domain is the set of x-values and the range is the set of y-values.

What you'll learn

Why it matters: Any model with practical limits has a restricted domain — you cannot have negative time or fractional people. Engineers and modelers track domain restrictions to avoid impossible inputs.

Worked example

Problem. Find the value of x that is excluded from the domain of f(x) = 1/(x - 4).

  1. The denominator x - 4 cannot equal zero.
  2. x - 4 = 0 means x = 4 is excluded.

Answer: 4

Practice problems

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: For f(x) = 1/(x - 4), enter the excluded x-value.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. x - 4 = 0 means x = 4 is excluded.
  4. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

2. For f(x) = sqrt(x - 3), enter the smallest valid x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For f(x) = sqrt(x - 3), enter the smallest valid x.
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. x - 3 must be >= 0.
  4. Smallest valid x is 3.
  5. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: For f(x) = 1/x, enter the excluded x-value.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. x cannot be 0 (division by zero).
  4. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: For f(x) = 1/(x + 5), enter the excluded x-value.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. x + 5 = 0 means x = -5 is excluded.
  4. Check the result by substituting or estimating: the response should match -5 and make sense in the original problem.

Answer: -5

5. For f(x) = x^2, enter the smallest possible output (the smallest value in the range).

Show solution
  1. Core Practice: First identify exactly what the question is asking: For f(x) = x^2, enter the smallest possible output (the smallest value in the range).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. x^2 is never negative.
  4. Smallest output is 0 at x = 0.
  5. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

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