Domain and Range

Learn how to identify the domain (input values) and range (output values) of a function.

Intermediate25 minLesson
Printable worksheet

Definition

The domain of a function is the set of all possible input values (x-values) that the function can accept.
The range of a function is the set of all possible output values (y-values) that the function can produce.
Think of a function like a machine:
  • Domain: What you can put INTO the machine
  • Range: What comes OUT of the machine

Try it now

What does the domain of a function represent?

Worked Examples

Find the domain of the function shown, where the graph exists from to .

1

Identify the leftmost x-value

The graph starts at → Left bound:

2

Identify the rightmost x-value

The graph ends at → Right bound:

3

Write in interval notation

All x-values from to , inclusive →

4

Write in set notation (alternative)

→ "x such that x is between -3 and 5"

Common Mistakes

Confusing domain with range

Why it's wrong: Both terms describe sets of values, but they apply to different variables.

Correct: Domain = x-values (horizontal, inputs). Range = y-values (vertical, outputs). Remember: "D" comes before "R" alphabetically, just like x comes before y.

Forgetting to exclude values that make denominators zero

Why it's wrong: Division by zero is undefined, so those x-values cannot be in the domain.

Correct: Always check if there's a variable in the denominator. Set denominator and solve.

Using wrong bracket notation

Why it's wrong: Square brackets include endpoints; parentheses exclude them.

Correct: means (includes a and b). means (excludes a and b). Always use with .

Assuming all functions have domain

Why it's wrong: Many functions have natural restrictions from fractions, square roots, or real-world contexts.

Correct: Check for: division by zero, square roots of negatives, logarithms of non-positives, and contextual limits.

Interactive Visual

Linear Function Explorer

y = x
Slope (m)1
Y-Intercept (b)0
b
run
rise
xy
-2-2
-1-1
00
11
22
to

Click on numbers to select them. Adjust the range to explore different values.

Interactive Sandbox

Interactive Grapher

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy

What does the domain of a function represent?

Why It Matters

Understanding domain and range helps you:
  • Avoid mathematical errors: You can't take the square root of a negative number (in real numbers), so has domain restrictions
  • Interpret real-world situations: A function modeling height vs. time has a range limited by physical constraints
  • Understand function behavior: Knowing the domain tells you where a function "lives" on the x-axis; the range tells you its vertical extent
  • Solve problems correctly: Many word problems require you to find realistic values within a function's domain and range

Real World Applications

Projectile Motion

When a ball is thrown, its height over time is a function with restricted domain and range.

Example:

If models height in meters after t seconds, the domain is (time from launch to landing) and range is (ground level to maximum height).

1Try It Yourself

A rocket's height is modeled by where t is in seconds.

What is the maximum height (top of the range)?

Step 1: Write the mathematical expression

The vertex occurs at :

Pricing Functions

Businesses use functions to model prices, with domains restricted to realistic quantities.

Example:

If gives the price per unit when selling x units, the domain might be (can't sell negative units, and price can't go below zero).

2Try It Yourself

A store's profit function is where x is items sold.

What's the realistic domain if profit must be non-negative?

Step 1: Write the mathematical expression

Solve

Key Takeaways

  • 1Domain is the set of all valid input values (x-values) for a function
  • 2Range is the set of all possible output values (y-values) a function can produce
  • 3Common domain restrictions: denominators , square root arguments
  • 4Use interval notation with for included endpoints and for excluded endpoints
  • 5Always use parentheses with infinity: or

Frequently Asked Questions

Use a bracket if the endpoint is included in the set (solid dot on graph, or in inequality). Use parenthesis if excluded (open dot, or ). Always use parentheses with infinity since infinity is not a number.
Use a bracket if the endpoint is included in the set (solid dot on graph, or in inequality). Use parenthesis if excluded (open dot, or ). Always use parentheses with infinity since infinity is not a number.
Then the domain is all real numbers: or . This is common for polynomials without fractions or roots, like or .
Consider what outputs are possible. For quadratics, find the vertex (minimum or maximum). For square roots, outputs are always . For rational functions, check horizontal asymptotes. Sometimes graphing is the easiest method!

Glossary

Domain
The set of all valid input values (x-values) for a function
Range
The set of all possible output values (y-values) of a function
Interval notation
A way to write sets using brackets and parentheses, e.g.,
Set-builder notation
A way to describe sets using conditions, e.g.,
Restriction
A value that cannot be in the domain due to mathematical rules

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