3.3Domain and Range
Every function has a set of valid inputs (domain) and a set of resulting outputs (range). Learn to identify both from ordered pairs, tables, mapping diagrams, and graphs.
Why This Matters
Domain and range define what inputs and outputs are valid for a function — critical for avoiding errors in science and engineering. You'll use these concepts constantly in Precalculus, Calculus, and data analysis.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
What information do the domain and range give us about a function, and why does it matter?
Lesson Overview
Every function has two important sets of values. The domain is the set of all valid input values — the x-values you are allowed to put into the function. The range is the set of all output values — the y-values the function actually produces. When a function is given as a list of ordered pairs, the domain is the set of all first coordinates and the range is the set of all second coordinates. We write these sets using set notation: curly braces with values listed and separated by commas, with no repeats.
Think of a function as a machine: you put in a domain value, it produces a range value.
f(x) = 2x + 1
Mapping diagrams show domain (left oval) and range (right oval) clearly.
Example A — Function
✓ FunctionExample B — Not a Function
✗ Not a FunctionExample A: Domain = {1, 2, 3, 4} · Range = {3, 5, 7, 9}
Example B: Input 2 maps to two outputs — not a function.
Domain and range can also be shown on number lines.
Domain (x-values)
Range (y-values)
Domain = inputs, Range = outputs
Mapping: domain → range
Worked Examples
Continuous domain: all real x values on the line
Discrete domain: only specific x values (dots)
Find the domain and range of: {(2, 5), (4, 9), (6, 13), (8, 17)}
List all first coordinates (x-values): 2, 4, 6, 8
These are the domain values — no repeats, so write them as a set.
List all second coordinates (y-values): 5, 9, 13, 17
These are the range values — no repeats.
Find the domain and range from the table: x: 1, 3, 5, 7 | y: 4, 4, 8, 10
Read the x-column: 1, 3, 5, 7 — no repeats.
Domain = {1, 3, 5, 7}
Read the y-column: 4, 4, 8, 10 — the value 4 appears twice.
In a set, we list each value only once.
Range = {4, 8, 10}
Find the domain and range from the mapping diagram: 2→6, 3→8, 4→6, 5→10.
Mapping: 2→6, 3→8, 4→6, 5→10
✓ FunctionDomain = inputs (left oval)
{2, 3, 4, 5}
Range = outputs (right oval)
{6, 8, 10}
Note: 6 appears twice as an output but is listed once in the range set.
The function f is defined by f(x) = 2x + 1 for x ∈ {0, 1, 2, 3}. Find the domain and range.
The domain is given: {0, 1, 2, 3}
Evaluate f for each domain value:
f(0) = 1, f(1) = 3, f(2) = 5, f(3) = 7
The range is the set of all outputs.
A school store sells pencils for $0.25 each. A student can buy 1–5 pencils. C(n) = 0.25n. Find the domain and range.
Domain = valid inputs = number of pencils: {1, 2, 3, 4, 5}
C(1) = $0.25, C(2) = $0.50, C(3) = $0.75, C(4) = $1.00, C(5) = $1.25
Guided Practice
Guided Practice Video: Domain and Range
Watch the guided practice walkthrough for domain and range, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Find the domain and range of: {(1, 7), (2, 9), (3, 11), (4, 13)}
Hint: The domain is the set of all x-values (first coordinates). The range is the set of all y-values (second coordinates). List each value only once.
Find the domain and range from the table: x: 0, 2, 4, 6 | y: 3, 3, 7, 9
Hint: Read the x-column for domain and the y-column for range. The value 3 appears twice in the y-column — write it only once in the range set.
A mapping diagram shows: 5→10, 6→12, 7→10, 8→14. Find the domain and range.
Hint: Domain = all inputs (left side of the diagram). Range = all outputs (right side). List 10 only once in the range.
The function g(x) = x² is defined for x ∈ {−2, −1, 0, 1, 2}. Find the domain and range.
Hint: The domain is given. Evaluate g(x) for each value: g(−2) = 4, g(−1) = 1, g(0) = 0, g(1) = 1, g(2) = 4. Then list the range values — no repeats.
A graph shows points at (−3, 2), (−1, 4), (0, 4), (2, −1). Find the domain and range.
Hint: Domain = all x-coordinates. Range = all y-coordinates. The value 4 appears twice — list it once.
f(x) = x + 1: domain all reals, range all reals
Key Vocabulary
Domain
The set of all valid input (x) values of a function.
Example: In {(1,5),(2,8),(3,11)}, the domain is {1,2,3}.
Range
The set of all output (y) values produced by a function.
Example: In {(1,5),(2,8),(3,11)}, the range is {5,8,11}.
Set Notation
A way to list elements of a set inside curly braces: {1, 2, 3}. No repeats; order doesn't matter.
Example: {3, 1, 2} and {1, 2, 3} are the same set.
Ordered Pair
A pair (x, y) where x is the input (domain value) and y is the output (range value).
Example: In (4, 9), the domain value is 4 and the range value is 9.
Discrete Function
A function with a countable, separate set of input/output values (like a table or list of points).
Example: {(1,2),(3,4),(5,6)} is a discrete function.
Mapping Diagram
A diagram with two ovals — inputs on the left, outputs on the right — connected by arrows.
Example: 1→3, 2→5, 3→7 shown with arrows between ovals.
Interactive Practice — 5 Questions
What is the domain of the relation {(3, 7), (5, 11), (7, 15)}?
What is the range of the relation {(1, 4), (2, 4), (3, 8), (4, 12)}?
A table shows: x = {2, 4, 6, 8}, y = {5, 9, 5, 13}. What is the range?
The function f(x) = 2x is defined for x ∈ {1, 2, 3}. What is the range?
A graph shows points at (−2, 5), (0, 3), (1, 5), (3, 7). What is the domain?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Find the domain and range: {(3, 8), (5, 12), (7, 16), (9, 20)}
Find the domain and range: {(−2, 5), (0, 3), (2, 5), (4, 1)}
Table: x = {2, 4, 6, 8}, y = {1, 3, 1, 5}. Find the domain and range.
f(x) = 3x − 1 for x ∈ {0, 1, 2, 3}. Find the domain and range.
A taxi charges $3 plus $2 per mile for 1–5 miles. Write the function, then find the domain and range.
Common Mistakes
Listing the range values in the wrong order — e.g., {5, 3, 1} instead of {1, 3, 5}.
Always list domain and range values in ascending (smallest to largest) order.
Confusing domain (inputs/x-values) with range (outputs/y-values).
Domain = all possible x-values (inputs). Range = all possible y-values (outputs).
Reading domain from a graph by looking at the y-axis instead of the x-axis.
Domain is read from the x-axis (horizontal). Range is read from the y-axis (vertical).
Repeating values in a set — e.g., writing {4, 4, 8} instead of {4, 8}.
In a set, each value is listed only once. If y = 4 appears three times, write 4 only once in the range.
Math Tips
Domain = x-values, Range = y-values: In an ordered pair (x, y), x is always domain and y is always range.
No repeats in a set: If x = 3 appears three times in a table, write 3 only once in the domain.
Order doesn't matter in a set: {1, 2, 3} and {3, 1, 2} are the same set — but listing in order is cleaner.
Range = outputs actually produced: The range is not all possible y-values — only the ones the function actually outputs.
From a graph: Domain = all x-coordinates of plotted points. Range = all y-coordinates of plotted points.