Domain and Range
Textbook Reference
| Primary source | OpenStax College Algebra 2e, Section 3.2: “Domain and Range” |
| Direct link | https://openstax.org/books/college-algebra-2e/pages/3-2-domain-and-range |
This source is free and openly licensed. College Algebra 2e Section 3.2 covers finding the domain and range from equations and graphs.
Key idea
The domain is the set of inputs a function is allowed to accept, and the range is the set of outputs it actually produces. Finding the domain is mostly a hunt for what is forbidden.
A function’s rule may break for certain inputs. Division by zero is forbidden, and the square root of a negative number is not a real value. The domain is everything the rule does not forbid. The range is the flip side: feed in every legal input and collect all the outputs that come out. Domain is about the inputs going in; range is about the outputs coming out.
This means: to find a domain, do not list what is allowed; find what is banned (zero denominators, negative radicands) and exclude exactly those values. Everything else is in.
Prerequisite Check
Before this lesson, make sure you can do all of the following:
If interval notation is unfamiliar, review Set-Builder and Interval Notation first.
Quick Reference
Definitions.
| Term | Meaning | Lives along |
|---|---|---|
| Domain | the set of all allowed input values | the $x$-axis (horizontal) |
| Range | the set of all output values produced | the $y$-axis (vertical) |
Finding the domain from an equation (the two banned moves).
| Watch for | Rule | Exclude |
|---|---|---|
| a denominator | it cannot be zero | the $x$ that makes it zero |
| an even root (square root, fourth root) | the inside cannot be negative | the $x$ that makes the inside negative |
If neither appears, the domain is all real numbers, $(-\infty, \infty)$.
The procedure (flashcard).
- Look for a denominator. Set it equal to zero and exclude those values.
- Look for an even root. Set the inside $\ge 0$ and solve; keep only those values.
- Write the surviving set in interval notation.
Key Concepts
1. Domain and Range as Inputs and Outputs
The domain of a function is the set of inputs for which the rule gives a real-number output. The range is the set of outputs the function actually produces as the input ranges over the domain.
Example 1. State the domain and range of $\{(1, 4), (2, 5), (3, 6)\}$.
The inputs are $1, 2, 3$, so the domain is $\{1, 2, 3\}$. The outputs are $4, 5, 6$, so the range is $\{4, 5, 6\}$.
For a function given by a finite list, the domain and range are just the input list and the output list.
2. Polynomials Have All Real Numbers as Domain
A polynomial such as $f(x) = 3x^2 - x + 1$ can be evaluated at any real number; there is no denominator and no even root to cause trouble. Its domain is all real numbers.
Example 2. State the domain of $f(x) = x^2 - 4x + 7$.
There is no denominator and no radical, so nothing is forbidden. The domain is $(-\infty, \infty)$.
3. Domain Restriction from a Denominator
A rational function has a variable in a denominator. Since division by zero is undefined, exclude every input that makes the denominator zero.
Example 3. Find the domain of $f(x) = \dfrac{1}{x - 3}$.
The denominator is zero when $x - 3 = 0$, that is, $x = 3$. Exclude $3$: \[ (-\infty, 3) \cup (3, \infty) \]
Example 4. Find the domain of $g(x) = \dfrac{x + 1}{x^2 - 4}$.
Set the denominator to zero: $x^2 - 4 = 0$, so $x = 2$ or $x = -2$. Exclude both: \[ (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \]
Important: only the denominator restricts the domain here. The numerator can be anything, including zero; a zero numerator just makes the output zero. It is the denominator that must avoid zero. Set the denominator, not the whole fraction, equal to zero.
4. Domain Restriction from an Even Root
A square root (or any even root) of a negative number is not a real number, so the expression under the root must be greater than or equal to zero.
Example 5. Find the domain of $f(x) = \sqrt{x - 2}$.
Require the inside to be non-negative: $x - 2 \ge 0$, so $x \ge 2$. The domain is \[ [2, \infty) \] The endpoint $2$ is included, because $\sqrt{0} = 0$ is a real number.
Important: even roots need the inside non-negative; odd roots do not. A square root requires a non-negative radicand, but a cube root accepts any real number (for example $\sqrt[3]{-8} = -2$). This restriction applies only to even roots.
5. Reading Domain and Range from a Graph
On a graph, the domain is the set of $x$-values the graph covers (its horizontal extent), and the range is the set of $y$-values the graph covers (its vertical extent).
Example 6. A parabola $f(x) = x^2$ opens upward with its lowest point at the origin. State its domain and range.
The graph extends left and right forever, so the domain is $(-\infty, \infty)$. The lowest output is $0$ (at the vertex), and outputs grow without bound, so the range is $[0, \infty)$.
Important: domain reads left-right, range reads up-down. Students sometimes report the range when asked for the domain. The domain is how wide the graph is (along $x$); the range is how tall it is (along $y$).
6. A Function with Two Restrictions
Example 7. Find the domain of $f(x) = \dfrac{\sqrt{x + 1}}{x - 2}$.
Two conditions must both hold:
- The radical requires $x + 1 \ge 0$, so $x \ge -1$.
- The denominator requires $x - 2 \neq 0$, so $x \neq 2$.
Combine: start with $x \ge -1$, then punch out $x = 2$: \[ [-1, 2) \cup (2, \infty) \]
Self-check: Find the domain of $f(x) = \dfrac{5}{x + 4}$.
The denominator is zero when $x + 4 = 0$, that is $x = -4$. Exclude it: $(-\infty, -4) \cup (-4, \infty)$.
Common Errors Summary
| Error | Example | Correction |
|---|---|---|
| Setting the numerator to zero for the domain | excluding $x = -1$ from $\frac{x+1}{x-3}$ | Only the DENOMINATOR restricts: exclude $x = 3$ |
| Requiring an odd root to be non-negative | restricting $\sqrt[3]{x}$ to $x \ge 0$ | Odd roots accept all reals; only even roots restrict |
| Excluding the endpoint of a square root | domain of $\sqrt{x-2}$ as $(2, \infty)$ | $\sqrt{0}$ is real, so include it: $[2, \infty)$ |
| Reporting range when asked for domain | giving $y$-values for the domain | Domain is the $x$-extent (horizontal) |
| Forgetting to combine two restrictions | using only the radical condition | Apply BOTH the radical and the denominator conditions |
Common Misconceptions
the domain is the same set as the range, or the two terms can be used interchangeably.
This is the input-output-confusion error. The domain is the set of allowed inputs (values of $x$), which lives along the horizontal axis. The range is the set of outputs actually produced (values of $y$), which lives along the vertical axis. For $f(x) = x^2$, the domain is $(-\infty, \infty)$ because any real number can be squared, but the range is $[0, \infty)$ because the output is always nonnegative. Reversing the two terms leads to incorrect statements about which values are allowed and which are produced.
Leveled Practice
Level 1 -- Direct Application
Problem 1. State the domain of $f(x) = 4x - 9$.
Show answer
A polynomial, no restrictions: $(-\infty, \infty)$.
Problem 2. State the domain of $f(x) = \dfrac{2}{x}$.
Show answer
Denominator zero at $x = 0$; exclude it: $(-\infty, 0) \cup (0, \infty)$.
Problem 3. State the domain of $f(x) = \sqrt{x}$.
Show answer
Require $x \ge 0$: $[0, \infty)$.
Level 2 -- Single Restriction
Problem 4. Find the domain of $f(x) = \dfrac{x}{x - 5}$.
Show answer
Denominator zero at $x = 5$; exclude it: $(-\infty, 5) \cup (5, \infty)$.
Problem 5. Find the domain of $g(x) = \sqrt{2x - 6}$.
Show answer
Require $2x - 6 \ge 0$, so $x \ge 3$: $[3, \infty)$.
Level 3 -- Combined and Graphical
Problem 6. Find the domain of $f(x) = \dfrac{\sqrt{x}}{x - 1}$.
Show answer
Radical requires $x \ge 0$; denominator requires $x \neq 1$. Combine: $[0, 1) \cup (1, \infty)$.
Problem 7. The graph of a function starts at the point $(-2, 0)$ (included) and rises forever to the right, never going below $y = 0$. State its domain and range.
Show answer
The graph covers $x$-values from $-2$ rightward, including $-2$: the domain is $[-2, \infty)$. The lowest output is $0$ (included) and outputs increase without bound: the range is $[0, \infty)$.
Mastery Checklist
You have mastered this skill when you can do all of the following without referring to notes:
Mental Model
Think of a function as a machine with an intake and an output chute.
The domain is the list of items the intake will accept without jamming. Two things jam the machine: an input that makes a denominator zero (the machine divides by nothing) and an input that puts a negative number under an even root (the machine cannot take that real-number square root). So to find the domain, you do not catalog the good inputs; you find the jams and rule out exactly those, and everything else is welcome. The range is whatever comes down the output chute once you have run every legal input through. Domain is the gate at the front; range is the pile at the back.
Connections
Within MATH 114
- Function notation and evaluation: the domain is exactly the set of inputs at which evaluation gives a real result.
- Rational and radical expressions: the same denominator-and-radical reasoning governs where those expressions are defined.
Toward Calculus (MATH 161)
- Continuity and limits: the points excluded from a domain (zero denominators in particular) are exactly where a function can fail to be continuous, the first thing calculus examines. Spotting domain restrictions now is the same skill as spotting discontinuities later.
Audience Notes
For students who find math intimidating: Finding a domain is detective work with only two suspects: a zero in a denominator and a negative under a square root. Check for those two, rule them out, and you are done. Everything not ruled out is allowed.
For students interested in proof: Stating a function’s domain precisely is part of defining the function at all; a rule without a domain is incomplete. This habit of specifying the domain is foundational in analysis, where the domain often carries as much meaning as the formula.
For students interested in careers: Input validation in software is the computing version of a domain: which inputs are legal, and which would crash the program. The reflex to ask “for which inputs does this break?” is valuable in any technical role.
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