Domain and Range Finder

Understanding Domain and Range

In algebra, functions are often visualized as machines: you put something in (input), and you get something out (output). Understanding **Domain** and **Range** is key to analyzing how these machines work.

What is the Domain?

The **Domain** is the set of all possible inputs (x-values) for which the function is defined. Think of it as "what am I allowed to plug into this equation?"

• For most simple equations like f(x) = 2x + 5, you can plug in any number. The domain is "All Real Numbers" or (-∞, ∞).
• However, some operations are illegal in math. For example, you cannot divide by zero, and you cannot take the square root of a negative number (in the real number system). These restrictions limit the domain.

What is the Range?

The **Range** is the set of all possible outputs (y-values) the function can produce. Think of it as "what acts can this machine perform?"

• For f(x) = x², no matter what number you square (positive or negative), the result is always positive or zero. You can never get -5 as an output. Therefore, the range is [0, ∞).

Common Function Examples

1. Linear Functions: f(x) = mx + b

Lines extends infinitely in both directions (unless vertical).
Domain: (-∞, ∞)
Range: (-∞, ∞)

2. Quadratic Functions: f(x) = ax² + bx + c

Parabolas open up or down. They cover all x-values, but the y-values are limited by the vertex.
Domain: (-∞, ∞)
Range: depends on vertex (e.g., [0, ∞) for basic x²)

3. Square Root Functions: f(x) = √x

You cannot have a negative number inside the root.
Domain: [0, ∞)
Range: [0, ∞)

4. Rational Functions: f(x) = 1/x

X cannot be zero because division by zero is undefined.
Domain: (-∞, 0) U (0, ∞)
Range: (-∞, 0) U (0, ∞)

Interval Notation Guide

Mathematicians use **Interval Notation** to write domains and ranges concisely:

• ( ) Parentheses: Indicate that the endpoint is NOT included (e.g., infinity or an undefined point).
• [ ] Brackets: Indicate that the endpoint IS included.
• U (Union symbol): Used to combine valid sections when there is a gap (like a hole in the graph).