Domain and Range
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn what domain and range mean, how to find them from a function rule or graph, and why they matter in real situations.
The Concept
Section titled “The Concept”- Domain: all possible input values (x-values) that a function can accept.
- Range: all possible output values (y-values) that the function can produce.
The diagram above shows how domain and range work. The green oval on the left holds the domain, the inputs we’re allowed to feed into the function. Those values pass through the function rule (here, f(x) = x² + 1), and the purple oval on the right collects the range, the outputs that come out. For example, putting in −2 gives (−2)² + 1 = 5, putting in 0 gives 1, and putting in 3 gives 10.
Notice that the domain usually covers a whole range of numbers, not just a few. In the diagram we only showed three sample inputs, but most functions accept infinitely many. For instance, f(x) = x² + 1 accepts every real number as input, so its domain is all real numbers. The range, however, is only y ≥ 1, because squaring always gives zero or more, the smallest output is 0² + 1 = 1. Different inputs can also produce the same output: both −2 and 2 give 5. That’s perfectly fine for a function. Multiple inputs can share an output, as long as each input gives exactly one output.
Number Sets and Notation
Section titled “Number Sets and Notation”When we say “all real numbers,” we mean every number on the number line: positive, negative, zero, fractions, decimals, irrational numbers like √2 and π. The symbol for the set of all real numbers is ℝ.
Here are the number sets you’ll see most often:
| Symbol | Name | Examples |
|---|---|---|
| ℕ | Natural numbers | 1, 2, 3, 4, … |
| ℤ | Integers | …, −2, −1, 0, 1, 2, … |
| ℚ | Rational numbers | 1/2, −3, 0.75, 7 |
| ℝ | Real numbers | all of the above plus √2, π, etc. |
Most functions in Algebra 2 work with ℝ unless there’s a specific restriction.
These symbols come from more formal mathematics and you’ll explore them in greater depth in Pre-Calculus and beyond. For now, the main one to know is ℝ. When a domain is “all real numbers,” you may see it written as ℝ or (−∞, ∞).
Interval Notation
Section titled “Interval Notation”Instead of writing “x ≥ 4” or “all real numbers except 3,” mathematicians use interval notation, a compact way to describe a set of numbers on the number line.
The key symbols:
- [ ] means the endpoint is included (solid dot on a number line).
- ( ) means the endpoint is excluded (open dot on a number line).
- ∞ and −∞ always get parentheses. You can never actually reach infinity.
Common patterns:
| In words | Inequality | Interval notation |
|---|---|---|
| All real numbers | −∞ < x < ∞ | (−∞, ∞) |
| x is at least 4 | x ≥ 4 | [4, ∞) |
| x is greater than 4 | x > 4 | (4, ∞) |
| x is between −1 and 5 (inclusive) | −1 ≤ x ≤ 5 | [−1, 5] |
| x is between −1 and 5 (exclusive) | −1 < x < 5 | (−1, 5) |
| All reals except 3 | x ≠ 3 | (−∞, 3) ∪ (3, ∞) |
The ∪ symbol means “union.” It joins two separate intervals together. You’ll use it whenever the domain has a gap (like a value that causes division by zero).
Set-Builder Notation
Section titled “Set-Builder Notation”You may also see set-builder notation, which reads like a sentence:
- {x | x ≥ 4} means “the set of all x such that x is greater than or equal to 4.”
- {x | x ≠ 3} means “the set of all x such that x is not equal to 3.”
The vertical bar ”|” is read as “such that.” Some textbooks use a colon ”:” instead.
Both interval notation and set-builder notation say the same thing. Interval notation is just more compact, and it’s the one you’ll use most in Algebra 2. Set-builder notation becomes more common in Pre-Calculus and college math, so it’s good to recognize it now even if you won’t use it every day yet.
For most functions we work with in Algebra 2:
- Domain restrictions usually come from:
- Division by zero (denominators cannot be zero)
- Square roots or even roots of negative numbers (cannot take the square root of a negative in real numbers)
- Range depends on the shape of the function (e.g., quadratics that open upward have a minimum value).
Examples:
- f(x) = 1 / (x − 3): domain is all real numbers except x = 3 (cannot divide by zero).
- g(x) = √x: domain is x ≥ 0 (cannot take the square root of a negative). Range is y ≥ 0.
When looking at a graph:
- Domain = all x-values the graph covers (left to right).
- Range = all y-values the graph covers (bottom to top).
Worked Example
Section titled “Worked Example”1. Find the domain of f(x) = 5 / (x + 2)
The denominator cannot be zero: x + 2 ≠ 0, so x ≠ −2.
Domain: all real numbers except −2.
2. Find the domain of g(x) = √(x − 4)
The expression inside the square root must be ≥ 0:
Domain: x ≥ 4.
3. Find the domain and range of h(x) = x² + 3
Domain: all real numbers (no restrictions, you can square any number).
Range: y ≥ 3 (the minimum value is 3, which occurs when x = 0).
Real-World Application
Section titled “Real-World Application”Domain and range have practical meaning:
- A phone plan charges 0.10 dollars per text: domain is number of texts ≥ 0 (you can’t send a negative number of texts).
- A rental car costs 45 dollars per day plus 0.25 dollars per mile: domain is miles ≥ 0.
- Temperature in Fahrenheit to Celsius: domain is all real numbers, but range depends on realistic temperatures.
- Profit function: range might start at a minimum loss and go up.
Understanding domain helps you avoid impossible inputs, and range shows you what outputs are possible.