Domain of Definition
What Is the Domain? – Introduction and Intuitive Understanding
🎯 What Is the Domain?
Domain = all x-values for which the function "works"
Every function has "rules of the game" — some x-values are allowed, others are not.
The domain is the set of all x-values that may be substituted into the function.
🌍 Real-Life Examples – Why Does It Matter?
🎢 Example 1: Roller coaster height
Suppose a function describes the height of a roller coaster during the ride.
What is the domain? Only the times when the ride is running!
You cannot ask "what is the height at minute 50?" if the ride lasts only 3 minutes.
Domain: \(0 \leq t \leq 3\) (minutes)
🌡️ Example 2: Freezer temperature
A function describing the freezer temperature throughout the day.
What is the domain? The 24 hours of the day!
There is no hour 25 or hour minus 3.
Domain: \(0 \leq t \leq 24\) (hours)
🍕 Example 3: Pizza price by size
A function computing the pizza price based on its diameter in cm.
What is the domain? Positive diameters only!
There is no pizza with a negative or zero diameter.
Domain: \(x > 0\) (cm)
👥 Example 4: Number of students in a class
A function computing the trip cost based on the number of students.
What is the domain? Positive integers only!
There are no 2.5 students or −3 students.
Domain: \(n \in \{1, 2, 3, 4, ...\}\)
⚠️ Why Is Finding the Domain Important?
1. To prevent mathematical errors
Some operations are forbidden in mathematics — without checking the domain we get wrong results!
2. To understand the graph
The domain determines where the graph exists — and where there are "holes" or jumps.
3. To solve word problems
In real-life problems, not every answer makes sense — the domain filters out unrealistic answers.
🚫 The Three Major Restrictions
In mathematics there are three operations that are simply forbidden::
🚫 Division by zero
\(\frac{?}{0}\)
Cannot divide by zero!
So the denominator: ≠ 0
🚫 Square root of a negative number
\(\sqrt{-?}\)
No square root of a negative number!
So under the root: ≥ 0
🚫 Log of a non-positive number
\(\log(-?)\)
No log of zero or a negative number!
So inside the log: > 0
📊 Types of Functions and Their Domains
| Function type | What to check? | Typical domain |
|---|---|---|
| Polynomial (like \(x^2+3x\)) | No restrictions! | ℝ (all real numbers) |
| Root function (like \(\sqrt{x-2}\)) | Under root ≥ 0 | Depends on the expression |
| Rational function (like \(\frac{1}{x-3}\)) | Denominator ≠ 0 | ℝ excluding points |
| Logarithm (like \(\log(x)\)) | Inside log > 0 | Positive only |
| Exponential (like \(2^x\)) | No restrictions! | ℝ (all real numbers) |
✏️ How to Write the Domain?
Two common methods:
Inequality notation
\(x \geq 3\)
\(x \neq 0\)
\(-2 < x < 5\)
Interval notation
\([3, \infty)\)
\(\mathbb{R} \setminus \{0\}\)
\((-2, 5)\)
💡 Notation reminder:
[ or ] = includes the endpoint (closed bracket)
( or ) = excludes the endpoint (open bracket)
∞ always with open bracket! (infinity is not a number)
📝 Summary
Domain of Definition = all x-values that may be substituted
Three restrictions: no division by 0, no root of negative, no log of ≤0
In the next pages we will learn to find the domain for each function type!