Domain and Range
Learn to express domain and range using interval and set notation, and determine them from equations and graphs.
Understanding Domain and Range
For any function y = f(x):
Domain
The set of all allowable x-values (inputs) for which the function is defined.
Think: "What values can I put into this function?"
Range
The set of all possible y-values (outputs) that the function can produce.
Think: "What values can come out of this function?"
Common restrictions on the domain include:
- Division by zero: The denominator cannot equal zero.
- Square roots: The expression under a square root must be ≥ 0.
- Logarithms: The argument of a logarithm must be > 0.
Notation for Domain and Range
There are two common ways to express domain and range:
| Interval Notation | Set Notation | Meaning |
|---|---|---|
| [a, b] | {x : a ≤ x ≤ b} | Closed interval — includes a and b |
| (a, b) | {x : a < x < b} | Open interval — excludes a and b |
| [a, b) | {x : a ≤ x < b} | Half-open — includes a, excludes b |
| (-∞, ∞) | {x : x ∈ ℝ} | All real numbers |
| [0, ∞) | {x : x ≥ 0} | Non-negative real numbers |
Key rule: Always use round brackets (parentheses) with ∞ and -∞ because infinity is not a number and cannot be included.
Reading Domain and Range from Graphs
To find the domain from a graph, look at the horizontal extent (left to right). To find the range, look at the vertical extent (bottom to top).
Common Functions and Their Domains/Ranges
y = x²
Domain: (-∞, ∞)
Range: [0, ∞)
y = √x
Domain: [0, ∞)
Range: [0, ∞)
y = 1/x
Domain: (-∞, 0) ∪ (0, ∞)
Range: (-∞, 0) ∪ (0, ∞)
y = x³
Domain: (-∞, ∞)
Range: (-∞, ∞)
Key Vocabulary
Domain
The complete set of input values (x-values) for which the function is defined.
Range
The complete set of output values (y-values) the function can produce.
Interval Notation
A way to express sets using brackets and parentheses, e.g. [2, 5) means 2 ≤ x < 5.
Set-Builder Notation
Notation using a condition to describe a set, e.g. {x : x ≥ 0} means all x where x is non-negative.
Worked Examples
Find the domain of f(x) = √(3x - 6).
Step 1: The expression under the square root must be ≥ 0: 3x - 6 ≥ 0.
Step 2: Solve: 3x ≥ 6, so x ≥ 2.
Answer: Domain = [2, ∞) or {x : x ≥ 2}.
Find the domain and range of f(x) = 1/x - 4.
Step 1: The denominator cannot be zero: x - 4 ≠ 0, so x ≠ 4.
Step 2: Domain = (-∞, 4) ∪ (4, ∞) or {x : x ≠ 4}.
Step 3: Since 1/(x-4) can never equal 0, the range excludes 0.
Answer: Domain = (-∞, 4) ∪ (4, ∞). Range = (-∞, 0) ∪ (0, ∞).
State the domain and range of f(x) = -(x + 1)² + 9.
Step 1: This is a downward-opening parabola (a = -1 < 0) with vertex at (-1, 9).
Step 2: Domain: all real numbers, (-∞, ∞).
Step 3: Since the parabola opens downward, the maximum y-value is 9.
Answer: Domain = (-∞, ∞). Range = (-∞, 9].
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the domain of f(x) = √(x - 5)?
Question 2
What is the range of f(x) = x² + 3?
Question 3
The interval [2, 7) means:
Question 4
What is the domain of f(x) = 1/(x² - 9)?
Question 5
What is the range of f(x) = -2(x - 3)² + 8?
Key Concepts Summary
- ●The domain is the set of all valid inputs (x-values); the range is the set of all possible outputs (y-values).
- ●Interval notation uses brackets: [ ] for included endpoints, ( ) for excluded endpoints.
- ●Always use round brackets with infinity since infinity is not a value.
- ●Check for division by zero, square root of negatives, and log of non-positives to find domain restrictions.
- ●The union symbol ∪ combines disconnected intervals, e.g. (-∞, 0) ∪ (0, ∞).