Every relationship has boundaries. In mathematics, we call relationships "functions," and their boundaries are defined by two key concepts: Domain and Range.
When you look at a function like a machine, the Domain describes everything you are allowed to put into the machine. The Range describes everything that can possibly come out of the machine. Understanding these concepts is essential for analyzing graphs, solving equations, and understanding real-world constraints.
1. What is the Domain?
The Domain is the set of all possible Input values (usually x) for which the function is defined.
[Image of domain and range mapping diagram]Think of it as the "Guest List" for a party. If a number is not on the list, it is not allowed in the function.
- Polynomials: Usually have a domain of "All Real Numbers" because you can plug any number into them.
- Fractions: You cannot divide by zero. So, if you have 1/x, x cannot be 0.
- Square Roots: You cannot take the square root of a negative number (in real numbers). So, the inside must be positive or zero.
2. What is the Range?
The Range is the set of all possible Output values (usually y or f(x)) that the function produces.
Think of this as the "Results." If you square a number (x2), the result is always positive or zero. You will never get a negative output. Therefore, the range is limited to non-negative numbers.
3. Visualizing on a Graph
The easiest way to find the domain and range is to look at the graph of the function.
[Image of domain and range on a graph]Finding Domain (Scan Left to Right)
Look at the x-axis. How far left does the graph go? How far right does it go?
- If there are arrows pointing left and right, the domain is likely Negative Infinity to Positive Infinity.
- If there is a hole or a break, that specific number is excluded.
Finding Range (Scan Bottom to Top)
Look at the y-axis. What is the lowest point? What is the highest point?
- If the graph is a "U" shape (parabola) sitting on the x-axis, the lowest y-value is 0, and it goes up forever.
4. Interval Notation
Mathematicians use a special shorthand called Interval Notation to write domain and range efficiently.
- Parentheses ( ): Use these when the number is NOT included (like < or >) or for Infinity (∞).
- Brackets [ ]: Use these when the number IS included (like ≤ or ≥).
Examples
| Inequality | Interval Notation | Meaning |
|---|---|---|
| x ≥ 5 | [5, ∞) | Starts at 5 (included) and goes up forever. |
| -3 < x ≤ 7 | (-3, 7] | Between -3 (excluded) and 7 (included). |
| All Real Numbers | (-∞, ∞) | No limits. |
5. Real-World Examples
Example 1: Time
If d(t) represents the distance you run over time t.
- Domain: Time cannot be negative. So the domain is t ≥ 0 or [0, ∞).
Example 2: A Square
If A(s) is the Area of a square with side length s.
- Domain: A side length must be positive. s > 0.
- Range: An area must also be positive. A > 0.
6. Conclusion
Domain and range are the "Terms and Conditions" of a function. The Domain tells you what is allowed to play, and the Range tells you what results you can expect. By checking for division by zero, square roots of negatives, and analyzing graphs, you can define these boundaries precisely.