Domain and Range
A free Algebra II lesson from the “Functions and Notation” 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 possible outputs. Restrictions can come from context, square roots, denominators, or a graph.
What you'll learn
- Identify domain and range
- Interpret restrictions
- Use interval and set descriptions
Worked example
Problem. A function is defined by the points (1, 5), (3, 7), and (6, 7). What is the range?
- The range is the set of y-values.
- The y-values are 5, 7, and 7.
- List repeats once: {5, 7}.
Answer: {5, 7}
Practice problems
1. For points (2, 8), (4, 9), and (6, 8), the domain is...
Choices: {2, 4, 6} · {8, 9} · {2, 8} · {4, 8, 9}
Show solution
- Warm-up: First identify exactly what the question is asking: For points (2, 8), (4, 9), and (6, 8), the domain is...
- For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
- Domain uses x-values.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: {2, 4, 6}
2. For points (2, 8), (4, 9), and (6, 8), the range is...
Choices: {8, 9} · {2, 4, 6} · {2, 8} · {4, 6}
Show solution
- Warm-up: First identify exactly what the question is asking: For points (2, 8), (4, 9), and (6, 8), the range is...
- For range questions, identify the possible output values after the input restrictions and graph shape are considered.
- Range uses y-values and removes repeats.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: {8, 9}
3. For f(x) = 1/(x - 5), what value is excluded from the domain?
Show solution
- Core Practice: First identify exactly what the question is asking: For f(x) = 1/(x - 5), what value is excluded from the domain?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- The denominator cannot be zero.
- x - 5 = 0 when x = 5.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
4. For g(x) = sqrt(x - 3), what is the smallest allowed x-value?
Show solution
- Core Practice: First identify exactly what the question is asking: For g(x) = sqrt(x - 3), what is the smallest allowed x-value?
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- The expression under the square root must be at least 0.
- Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.
Answer: 3
5. A movie ticket price function uses number of tickets as the input. Which domain makes sense?
Choices: Whole numbers 0 or greater · All real numbers · Only negative numbers · Only decimals between 0 and 1
Show solution
- Challenge: First identify exactly what the question is asking: A movie ticket price function uses number of tickets as the input. Which domain makes sense?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- You cannot buy a negative or fractional number of tickets.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Whole numbers 0 or greater
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