Quick Answer
The domain is the set of all valid inputs (x-values). The range is the set of all possible outputs (y-values). From a graph, read left-to-right for domain and bottom-to-top for range. From an equation, find any x-values that cause division by zero or a negative under a square root — those are excluded from the domain.
Domain vs. Range — The Core Idea
Think of a function as a machine. You feed it an input (x), it produces an output (y = f(x)).
Domain: the set of all inputs the machine will accept without breaking.
Range: the set of all outputs the machine can possibly produce.
In interval notation, you write domains and ranges as intervals: [a, b] means a ≤ x ≤ b (closed, includes endpoints); (a, b) means a < x < b (open, excludes endpoints). Use ∞ for unbounded intervals, always with a parenthesis: (−∞, ∞).
Finding Domain and Range from a Graph
Domain from a graph:
Look at the graph from left to right. The domain is the set of all x-values the graph covers.
• If the graph extends infinitely left and right, the domain is (−∞, ∞).
• If the graph starts at x = 2 with a closed dot, the domain starts at [2, …).
• An open dot at x = 5 means x = 5 is NOT included: use a parenthesis.
Range from a graph:
Look at the graph from bottom to top. The range is the set of all y-values the graph reaches.
• If the lowest point is y = −3 (closed dot), the range starts at [−3, …).
• If the graph has a horizontal asymptote at y = 0 that it never touches, the range excludes 0.
Finding Domain from an Equation
Two main restrictions to check:
1. Division by zero: any x-value that makes the denominator equal zero is excluded.
Example: f(x) = 1/(x − 3). Set x − 3 = 0 → x = 3. Domain: (−∞, 3) ∪ (3, ∞).
2. Even roots of negatives: any x-value that makes the expression under an even root negative is excluded.
Example: f(x) = √(x − 4). Set x − 4 ≥ 0 → x ≥ 4. Domain: [4, ∞).
If neither restriction applies (polynomial, exponential, sine/cosine), the domain is all real numbers: (−∞, ∞).
Finding Range from an Equation
Range is harder to find algebraically. Common approaches:
1. Know the standard shapes:
• Linear f(x) = mx + b: range is (−∞, ∞).
• Quadratic f(x) = ax² + bx + c: range is [vertex y-value, ∞) if a > 0, or (−∞, vertex y-value] if a < 0.
• Square root f(x) = √x: range is [0, ∞).
• Absolute value f(x) = |x|: range is [0, ∞).
2. Solve for x in terms of y, then find which y-values allow a valid x.
3. Use a graph (sketch or calculator) to read the range visually.
Finding Domain and Range from a Table
From a table of (x, y) pairs:
Domain: list all the x-values in the table.
Range: list all the y-values in the table.
If the table represents a discrete function (only those specific points), the domain and range are just those listed values — written as sets: {1, 3, 5, 7}.
If the table samples a continuous function, you need more information (the equation or graph) to state the full domain and range.
Worked Examples
Example 1
Find the domain and range of f(x) = √(2x − 6).
Solution
- Domain: the expression under the square root must be ≥ 0.
- 2x − 6 ≥ 0 → 2x ≥ 6 → x ≥ 3.
- Domain: [3, ∞).
- Range: √(anything ≥ 0) ≥ 0. The minimum output is √0 = 0 (when x = 3). As x → ∞, f(x) → ∞.
- Range: [0, ∞).
Answer: Domain: [3, ∞). Range: [0, ∞).
Example 2
Find the domain of f(x) = (x + 1) / (x² − x − 6).
Solution
- Exclude x-values that make the denominator zero.
- Factor: x² − x − 6 = (x − 3)(x + 2).
- Set each factor to zero: x = 3 or x = −2.
- Domain: all real numbers except x = 3 and x = −2.
- In interval notation: (−∞, −2) ∪ (−2, 3) ∪ (3, ∞).
Answer: Domain: (−∞, −2) ∪ (−2, 3) ∪ (3, ∞).
Example 3
Find the domain and range of f(x) = −x² + 4.
Solution
- Domain: f(x) is a polynomial — no restrictions. Domain: (−∞, ∞).
- Range: this is a downward-opening parabola (a = −1 < 0).
- Vertex is at x = 0: f(0) = 4. This is the maximum output.
- As x → ±∞, f(x) → −∞.
- Range: (−∞, 4].
Answer: Domain: (−∞, ∞). Range: (−∞, 4].
Common Mistakes
Confusing domain and range. Domain = x-values (inputs, horizontal axis). Range = y-values (outputs, vertical axis).
Using the wrong bracket type. A closed dot or ≤ uses a square bracket [. An open dot or < uses a parenthesis (. Infinity always uses a parenthesis.
Forgetting to check both restrictions (division by zero AND square root of negative). A function can have both at once.
Stating the range of a quadratic without finding the vertex first. The vertex gives the minimum or maximum y-value.
Practice Problems
Find the domain of f(x) = 1/(x + 5).
Hint: What x-value makes the denominator zero?
Find the domain and range of f(x) = √(x + 9).
Hint: Set x + 9 ≥ 0 for domain. What is the minimum output?
Find the domain and range of f(x) = x² − 2x + 1.
Hint: Complete the square or find the vertex to get the range.
A table shows: x = {−2, 0, 3, 5}, y = {4, 4, 1, 7}. State the domain and range.
Hint: List the x-values for domain, y-values for range. Are any repeated?
Frequently Asked Questions
What is the domain and range of a function?
The domain is the set of all valid input values (x-values) a function can accept. The range is the set of all possible output values (y-values) the function can produce.
How do you find the domain of a function?
Check for two restrictions: (1) values that make a denominator equal zero — exclude those; (2) values that make an expression under an even root negative — exclude those. If neither restriction applies (e.g., a polynomial), the domain is all real numbers (−∞, ∞).
How do you find the range of a function?
For common function types: a linear function has range (−∞, ∞); a quadratic has range [vertex y-value, ∞) if it opens up or (−∞, vertex y-value] if it opens down; a square root function has range [0, ∞). For other functions, solve for x in terms of y and determine which y-values allow a valid x.
What is interval notation for domain and range?
Interval notation uses brackets and parentheses: [a, b] means a ≤ x ≤ b (closed, includes endpoints); (a, b) means a < x < b (open, excludes endpoints). Infinity is always written with a parenthesis: (−∞, ∞). Use ∪ to join separate intervals, e.g. (−∞, 3) ∪ (3, ∞).
What is the difference between domain and range?
Domain refers to inputs (x-values, the horizontal axis). Range refers to outputs (y-values, the vertical axis). A common memory trick: Domain = x, Range = y; D comes before R alphabetically, just as x comes before y.
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