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
Domain→functionRange\text{Domain} \xrightarrow{\text{function}} \text{Range}
x→ff(x)=yx \xrightarrow{f} f(x) = y

Try it now

What does the domain of a function represent?

Worked Examples

Find the domain of the function shown: its graph starts at a solid dot at x=−3x = -3 and ends at a solid dot at x=5x = 5.
1

Identify the leftmost x-value

The graph starts at x=−3x = -3 → Left bound: −3-3
2

Identify the rightmost x-value

The graph ends at x=5x = 5 → Right bound: 55
3

Write in interval notation

All x-values from −3-3 to 55, inclusive → [−3,5][-3, 5]
4

Write in set notation (alternative)

{x∣−3≤x≤5}\{x \mid -3 \leq x \leq 5\} → "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 ≠0\neq 0 and solve.

Using wrong bracket notation

Why it's wrong: Square brackets [][ ] include endpoints; parentheses ()( ) exclude them.

Correct: [a,b][a, b] means a≤x≤ba \leq x \leq b (includes a and b). (a,b)(a, b) means a<x<ba < x < b (excludes a and b). Always use ()( ) with ∞\infty.

Assuming all functions have domain (−∞,∞)(-\infty, \infty)

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.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Find the Domain of a Function | Step-by-Step Guide & Examples

by Understand The Math

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Interactive Visual

Function graph

-4-2024-202468xy(0, -1)(-3, 0)

Tap or hover over the graph to read off values.

f(x) = x² − 1g(x) = √(x + 3)
  • Vertex: (0, -1)
to

x < 2 or x > 2·(−∞, 2) ∪ (2, ∞)

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

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

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x^2 - 1

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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 f(x)=xf(x) = \sqrt{x} 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 h(t)=−5t2+20th(t) = -5t^2 + 20t models height in meters after t seconds, the domain is [0,4][0, 4] (time from launch to landing) and range is [0,20][0, 20] (ground level to maximum height).

1Try It Yourself

A rocket's height is modeled by h(t)=−4t2+32th(t) = -4t^2 + 32t where t is in seconds.

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

Step 1: Write the mathematical expression

Find the vertex time t=−b2at = \frac{-b}{2a}, then the height hh at that time:

Pricing Functions

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

Example:

If P(x)=100−2xP(x) = 100 - 2x gives the price per unit when selling x units, the domain might be [0,50][0, 50] (can't sell negative units, and price can't go below zero).

2Try It Yourself

A store's profit function is P(x)=−x2+50x−400P(x) = -x^2 + 50x - 400, where x is the number of items sold.

The store does not want to make a loss, so P(x)≥0P(x) \geq 0. What is the largest number of items in this realistic domain?

Step 1: Write the mathematical expression

Solve −x2+50x−400≥0-x^2 + 50x - 400 \geq 0 and give the right endpoint:

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 ≠0\neq 0, square root arguments ≥0\geq 0
  • 4Use interval notation with [][ ] for included endpoints and ()( ) for excluded endpoints
  • 5Always use parentheses with infinity: (−∞,a](- \infty, a] or [b,∞)[b, \infty)

Frequently Asked Questions

Use a bracket [][ ] if the endpoint is included in the set (solid dot on graph, ≤\leq or ≥\geq 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, ≤\leq or ≥\geq 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: (−∞,∞)(-\infty, \infty) or R\mathbb{R}. This is common for polynomials without fractions or roots, like f(x)=2x+3f(x) = 2x + 3 or g(x)=x3−xg(x) = x^3 - x.
Consider what outputs are possible. For quadratics, find the vertex (minimum or maximum). For square roots, outputs are always ≥0\geq 0. 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., [−3,5)[-3, 5)
Set-builder notation
A way to describe sets using conditions, e.g., {x∣x>0}\{x \mid x > 0\}
Restriction
A value that cannot be in the domain due to mathematical rules
Today's challengeOne problem, three tries

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