Domain and Range
A free College Algebra lesson from the “Functions” unit, with a worked example and practice problems including step-by-step solutions.
Domain is the set of allowed inputs, and range is the set of possible outputs. Restrictions may come from denominators, radicals, graphs, or real-world meaning.
What you'll learn
- Find domain and range from representations
- Identify restrictions
- Use context to limit domains
Worked example
Problem. For f(x) = 1/(x - 3), what value is excluded from the domain?
- The denominator cannot equal zero.
- x - 3 = 0 when x = 3.
- So x = 3 is excluded.
Answer: 3
Practice problems
1. For points (1, 5), (2, 8), and (4, 8), the range is...
Choices: {5, 8} · {1, 2, 4} · {1, 5} · {2, 4, 8}
Show solution
- Warm-up: First identify exactly what the question is asking: For points (1, 5), (2, 8), and (4, 8), the range is...
- For range questions, identify the possible output values after the input restrictions and graph shape are considered.
- Range uses output values.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: {5, 8}
2. For f(x) = 1/(x + 9), what value is excluded?
Show solution
- Core Practice: First identify exactly what the question is asking: For f(x) = 1/(x + 9), what value is excluded?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- x + 9 cannot be 0.
- Check the result by substituting or estimating: the response should match -9 and make sense in the original problem.
Answer: -9
3. For g(x) = sqrt(x - 7), what is the smallest allowed x-value?
Show solution
- Challenge: First identify exactly what the question is asking: For g(x) = sqrt(x - 7), what is the smallest allowed x-value?
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- x - 7 must be at least 0.
- Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.
Answer: 7
4. For f(x) = 1/(x - 5), what value is excluded?
Show solution
- Restrictions: First identify exactly what the question is asking: For f(x) = 1/(x - 5), what value is excluded?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- The denominator cannot be zero.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
5. For g(x) = sqrt(12 - x), what is the largest allowed x-value?
Show solution
- Radicals: First identify exactly what the question is asking: For g(x) = sqrt(12 - x), what is the largest allowed x-value?
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- 12 - x must be at least 0.
- Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.
Answer: 12
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