What Is A Domain And Range Of A Graph ?

If you’re learning algebra, functions, or graphs, one concept shows up everywhere:

What is a domain and range of a graph?

Once you truly understand this, reading graphs becomes much easier. This guide explains domain and range in clear, simple language, with practical steps you can use in homework, exams, or real-life problems.

Quick Definition (Plain English)

• Domain: all the x-values you are allowed to use
• Range: all the y-values you get after using those x-values

In short:

Domain goes in. Range comes out.

Why Domain and Range Matter

Domain and range help you:

• Understand what a graph represents
• Avoid impossible values
• Solve equations correctly
• Read real-world data accurately
• Prevent math mistakes on exams

Almost every function has rules about which values make sense.

Domain and Range on a Graph (Visual Meaning)

When you look at a graph:

• Domain = how far the graph stretches left to right
• Range = how far the graph stretches down to up

Think of it like this:

• Scan the graph horizontally → domain
• Scan the graph vertically → range

Example 1: Simple Line Graph

Imagine the graph of:

y = x + 2

Domain

• You can plug in any real number for x
• The graph extends forever left and right

Domain: all real numbers
(−∞, ∞)

Range

• The output also goes forever up and down

Range: all real numbers
(−∞, ∞)

Example 2: Quadratic Graph (Parabola)

Consider:

y = x²

Domain

• You can square any number

Domain: all real numbers

Range

• Squares are never negative
• The lowest value is 0

Range: y ≥ 0
[0, ∞)

How to Find the Domain of a Graph (Step-by-Step)

Step 1: Look for Breaks or Gaps

• Holes
• Open circles
• Vertical asymptotes

These usually mean certain x-values are not allowed.

Step 2: Watch for Restrictions

Some expressions create limits:

Situation	What to Avoid
Division	Cannot divide by 0
Square root	Cannot be negative (in basic math)
Logarithm	Must be greater than 0

Step 3: Read the Graph Left to Right

Ask:

“Which x-values appear anywhere on the graph?”

That set is your domain.

How to Find the Range of a Graph (Step-by-Step)

Step 1: Look at Lowest and Highest Points

• Minimum value
• Maximum value

Step 2: Check Direction

• Does the graph go up forever?
• Does it stop at a point?

Step 3: Read the Graph Bottom to Top

Ask:

“Which y-values does the graph actually reach?”

That set is your range.

Example 3: Square Root Graph

y = √x

Domain

• You cannot take the square root of a negative number

Domain: x ≥ 0
[0, ∞)

Range

• Square roots are never negative

Range: y ≥ 0
[0, ∞)

Example 4: Rational Function

y = 1 / x

Domain

• Division by zero is not allowed

Domain: x ≠ 0
(−∞, 0) ∪ (0, ∞)

Range

• The function never equals 0

Range: y ≠ 0
(−∞, 0) ∪ (0, ∞)

Domain and Range Using Interval Notation

You’ll often see answers written like this:

• ( ) means the number is not included
• [ ] means the number is included

Examples:

• x > 2 → (2, ∞)
• x ≥ −1 → [−1, ∞)
• −3 ≤ x < 4 → [−3, 4)

Domain and Range of Discrete Graphs

Some graphs don’t connect smoothly.

Example: a graph with dots only

Domain

• Only the x-values where dots exist

Range

• Only the y-values where dots exist

No guessing between points.

Real-Life Example of Domain and Range

Phone Data Plan

• Domain: number of gigabytes used (0 or more)
• Range: total cost (cannot be negative)

Height Over Time

• Domain: time (starts at 0)
• Range: height (limited by reality)

Domain and range help keep answers realistic.

Common Mistakes Students Make

❌ Mixing up domain and range
❌ Forgetting restrictions
❌ Including values that aren’t on the graph
❌ Ignoring open circles
❌ Assuming graphs go forever

Always check the picture, not just the equation.

Domain vs Range (Quick Comparison)

Feature	Domain	Range
Axis	x-axis	y-axis
Direction	Left → Right	Down → Up
Meaning	Inputs	Outputs
Question	What can x be?	What can y be?

How Teachers Expect Answers

Depending on the problem, you may need:

• Interval notation
• Inequality form
• Set-builder notation
• Written explanation

Always read the question carefully.

Final Summary

So, what is a domain and range of a graph?

• Domain is all possible x-values shown on the graph
• Range is all possible y-values the graph produces
• You find domain by reading left to right
• You find range by reading bottom to top