1.2Identifying Domains and Ranges
Find the domain and range of functions from equations, graphs, and tables. Write answers in interval notation and set-builder notation.
Domain and range define the boundaries of a function. Knowing what inputs are valid and what outputs are possible is essential for solving equations, graphing, and avoiding undefined expressions in calculus.
Essential Question
How do we determine which input values are valid for a function, and what output values it can produce?
Lesson Overview
The domain of a function is the set of all valid input values (x-values). For most functions, the domain is all real numbers unless there is a square root of a negative number or division by zero — both of which are undefined in the real number system.
The range of a function is the set of all possible output values (y-values). The range is determined by the behavior of the function — for example, since x² is always non-negative, the range of f(x) = x² is [0, ∞).
We express domains and ranges using interval notation, which uses parentheses for open (excluded) endpoints and brackets for closed (included) endpoints. For example, [3, ∞) means all numbers greater than or equal to 3. We can also use set-builder notation: { x | x ≥ 3 }, read "the set of all x such that x is greater than or equal to 3."
Visual Reference
Interval Notation Reference
f(x) = x² — Domain & Range
Common Domain Restrictions
| Function Type | Restriction | Example | Domain |
|---|---|---|---|
| Square root | Radicand ≥ 0 | √(x − 3) | [3, ∞) |
| Rational | Denominator ≠ 0 | 1/(x − 2) | (−∞,2)∪(2,∞) |
| Even root | Radicand ≥ 0 | ⁴√(x + 1) | [−1, ∞) |
| Logarithm | Argument > 0 | log(x + 4) | (−4, ∞) |
Worked Examples
Find the domain of f(x) = √(x − 4).
The radicand must be ≥ 0: x − 4 ≥ 0
Solve: x ≥ 4
Find the domain of g(x) = 3/(x² − 9).
Denominator cannot equal zero: x² − 9 ≠ 0
Factor: (x − 3)(x + 3) ≠ 0 → x ≠ 3 and x ≠ −3
Find the domain and range of h(x) = x² + 1.
Domain: x² is defined for all real numbers — no restrictions.
Range: x² ≥ 0 for all x, so x² + 1 ≥ 1.
The minimum output is 1, achieved at x = 0.
Find the domain of f(x) = √(2x + 6) / (x − 1).
Radicand ≥ 0: 2x + 6 ≥ 0 → x ≥ −3
Denominator ≠ 0: x − 1 ≠ 0 → x ≠ 1
Combine: x ≥ −3 AND x ≠ 1
A parabola has vertex at (2, −3) and opens upward, defined for all x. Find the domain and range.
Domain: the parabola is defined for every x-value — no restrictions.
Range: the vertex is the minimum point at y = −3; the parabola opens upward.
Guided Practice
External Supplemental Resource
Guided Practice Video: Identifying Domains and Ranges
Review how to identify the domain and range of a function from its rule and graph before completing the guided practice problems.
Video by Khan Academy on YouTube
Watch on YouTube ↗Find the domain of f(x) = √(x + 7). Write in interval notation.
Hint: Set x + 7 ≥ 0 and solve for x.
Find the domain of g(x) = 5/(x + 4). Write in interval notation.
Hint: What value of x makes the denominator zero?
Find the domain of h(x) = √(3 − x). Write in interval notation.
Hint: 3 − x ≥ 0 means x ≤ 3. Which direction does the interval go?
Find the domain and range of f(x) = |x| − 2. Write both in interval notation.
Hint: |x| ≥ 0 for all x, so what is the minimum value of |x| − 2?
Find the domain of f(x) = 1/√(x − 5). Write in interval notation.
Hint: The square root is in the denominator — you need x − 5 > 0 (strictly), not ≥ 0.
Key Vocabulary
Domain
The set of all valid input values (x-values) for a function.
Example: For f(x) = √x, the domain is [0, ∞).
Range
The set of all possible output values (y-values) of a function.
Example: For f(x) = x², the range is [0, ∞).
Interval notation
A way to write a set of numbers using brackets [ ] (inclusive) and parentheses ( ) (exclusive).
Example: [2, 5) means 2 ≤ x < 5.
Set-builder notation
Written as { x | condition }, read "the set of all x such that condition."
Example: {x | x ≥ 3} is the same as [3, ∞).
Restriction
A value excluded from the domain because it causes division by zero or a square root of a negative number.
Example: In g(x) = 1/(x−2), x = 2 is a restriction.
Natural domain
The largest possible domain for a function — all real numbers that produce a defined output.
Example: The natural domain of f(x) = x² is (−∞, ∞).
Check Your Understanding
Interactive Practice — 5 Questions
What is the domain of f(x) = √(x − 6)?
What is the domain of g(x) = 4/(x − 3)?
What is the range of f(x) = x² − 5?
Which interval notation represents x ≥ −2?
What is the domain of h(x) = √(4 − x²)?
Common Mistakes
Writing domain as (4, ∞) for f(x) = √(x − 4) — using a parenthesis instead of a bracket.
x = 4 makes the radicand exactly 0, which is valid (√0 = 0). Use a bracket: [4, ∞).
Thinking 1/√(x − 5) only needs x − 5 ≥ 0.
When the square root is in the denominator, the radicand must be strictly positive (x − 5 > 0) because √0 = 0 causes division by zero. Domain: (5, ∞).
Assuming x² has the same domain restriction as √x.
x² is defined for ALL real numbers. Only even roots and denominators create domain restrictions.
Writing range as (−∞, ∞) for f(x) = x² + 3.
x² is always ≥ 0, so x² + 3 ≥ 3. The range is [3, ∞), not all real numbers.
Math Tips
Domain checklist: (1) Is there a square root? Set radicand ≥ 0. (2) Is there a fraction? Set denominator ≠ 0. (3) Is there a log? Set argument > 0. If none of these apply, domain = (−∞, ∞).
To find range, ask: what is the minimum (or maximum) output? For f(x) = x² + k, the range is [k, ∞) because x² ≥ 0.
Interval notation quick rule: use [ ] when the endpoint IS included, ( ) when it is NOT. Infinity always gets ( ) because it is never actually reached.
SAT/ACT tip: domain questions often involve √(expression) or 1/(expression). When a function has both, check both restrictions and combine carefully.