Back to Lesson

Teacher Guide: Domain and Range

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

Present LessonShare to ClassroomView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Functions. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define domain and range in the context of functions
  • Identify domain and range from graphs of functions
  • Determine domain restrictions from function equations
  • Express domain and range using interval and set-builder notation
  • Apply domain and range concepts to real-world problems
Prerequisites
  • • Understanding of what a function is
  • • Familiarity with coordinate graphing
  • • Basic inequality solving
  • • Knowledge of function notation f(x)f(x)
Discussion Starters
  • 1. Why can't we take the square root of a negative number (in real numbers)? What would it mean?
  • 2. Can a function have a domain that's a single point? What would that look like?
  • 3. How would you explain domain and range to someone using a vending machine analogy?
  • 4. What real-world situations have natural domain restrictions?
Common Misconceptions

The domain must always start at zero

Remediation: Show functions with domains like (−∞,∞)(-\infty, \infty) or [−5,3][-5, 3]. The domain depends on the function, not a universal rule.

Range is just the height of the graph at one point

Remediation: Emphasize range is ALL possible y-values. Use vertical scanning: 'What y-values does this graph ever touch?'

If a number isn't in the domain, the function equals zero there

Remediation: Clarify that undefined means the function doesn't exist at that point - it's not zero, it's simply not defined.

Differentiation Ideas

For Struggling Students:

  • • Focus on reading domain/range from graphs before equations
  • • Use color coding: blue for domain (horizontal), red for range (vertical)
  • • Start with linear functions that have no restrictions

For On-Level Students:

  • • Find domain and range from various function types
  • • Practice converting between interval and inequality notation
  • • Identify restrictions in rational and radical functions

For Advanced Students:

  • • Explore piecewise functions with multiple domain pieces
  • • Find domain and range of composite functions
  • • Investigate how transformations affect domain and range
Standards Alignment
  • F.IF.A.1 (CCSS.MATH.CONTENT.HSF.IF.A.1)

    Understand that a function from one set (domain) to another set (range) assigns to each element of the domain exactly one element of the range

  • F.IF.B.5 (CCSS.MATH.CONTENT.HSF.IF.B.5)

    Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes

Lesson Resources
  • visualInteractive Function Explorer

    Students manipulate functions and observe how domain and range change

  • activityDomain Detective

    Find restrictions in various function types

  • worksheetInterval Notation Practice

    Convert between graphical, interval, and set-builder representations

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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

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.

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

How do I know if an endpoint should use a bracket or parenthesis?

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.

What if a function has no restrictions?

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.

How do I find the range without graphing?

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

More in This Topic

Function Notation f(x)

Intermediate25 min