What are Domain and Range?
The domain and range are fundamental concepts in mathematics that describe the input and output values of a function. Understanding domain and range is essential for analyzing functions, solving equations, and working with interval notation.
Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. It represents all the values you can "put into" the function.
Think of domain as the "allowed inputs" or the x-values that make the function work.
Range
The range of a function is the set of all possible output values (y-values) that the function can produce. It represents all the values you can "get out of" the function.
Think of range as the "possible outputs" or the y-values the function can produce.
How to Find the Domain of a Function
Finding the domain requires identifying values that would make the function undefined or invalid. This often involves solving compound inequalities to determine valid input ranges. Common restrictions include:
Domain Restrictions:
- 1.Division by zero: Values that make the denominator zero are excluded
- 2.Square roots of negative numbers: Expressions under even roots must be ≥ 0
- 3.Logarithms: Arguments must be greater than zero
- 4.Real-world constraints: Context may limit domain (e.g., negative time doesn't make sense)
Step-by-Step Method to Find Domain
- Identify the type of function (polynomial, rational, radical, etc.)
- Look for values that would make the function undefined
- Exclude those values from the domain
- Express the domain using interval notation or set notation
Domain and Range Examples
Example 1: Polynomial Function
Function: f(x) = x² + 3x - 2
Domain: All real numbers (no restrictions)
In interval notation: (-∞, ∞)
Range: Since it's a parabola opening upward with vertex at x = -3/2, the range is all y ≥ f(-3/2)
Range: [-17/4, ∞)
Example 2: Rational Function
Function: f(x) = 1/(x - 2)
Domain: All real numbers except x = 2 (division by zero)
In interval notation: (-∞, 2) ∪ (2, ∞)
Range: All real numbers except y = 0
Range: (-∞, 0) ∪ (0, ∞)
Example 3: Square Root Function
Function: f(x) = √(x - 3)
Domain: x - 3 ≥ 0, so x ≥ 3
In interval notation: [3, ∞)
Range: Since square root is always non-negative: y ≥ 0
Range: [0, ∞)
Example 4: Absolute Value Function
Function: f(x) = |x - 1| + 2
Domain: All real numbers
In interval notation: (-∞, ∞)
Range: Since |x - 1| ≥ 0, we have f(x) ≥ 2
Range: [2, ∞)
How to Find the Range of a Function
Finding the range is often more challenging than finding the domain. Sometimes this involves solving compound inequalities to determine the output values. Here are common methods:
Graphical Method
Graph the function and identify the y-values that the graph covers. This is the most visual approach.
Algebraic Method
Solve for x in terms of y and determine which y-values produce real x-values. This works well for many functions.
Calculus Method
Use derivatives to find maximum and minimum values, which help determine the range boundaries.
Function Properties
Use known properties (e.g., square roots are non-negative, absolute values are non-negative) to determine range.
Domain and Range in Interval Notation
Interval notation is the preferred way to express domain and range because it's concise and clear. Here's a quick reference:
| Description | Interval Notation | Example |
|---|---|---|
| All real numbers | (-∞, ∞) | Polynomial functions |
| Greater than or equal to a | [a, ∞) | Square root functions |
| Less than or equal to b | (-∞, b] | Some exponential functions |
| Between a and b (inclusive) | [a, b] | Restricted functions |
| All except c | (-∞, c) ∪ (c, ∞) | Rational functions |
Common Function Types and Their Domain/Range
Linear Functions: f(x) = mx + b
(-∞, ∞)(-∞, ∞) (if m ≠ 0)Quadratic Functions: f(x) = ax² + bx + c
(-∞, ∞)[k, ∞), if a < 0: (-∞, k]Exponential Functions: f(x) = aˣ
(-∞, ∞)(0, ∞)Logarithmic Functions: f(x) = logₐ(x)
(0, ∞)(-∞, ∞)Practice Problems
Problem 1
Find the domain and range of f(x) = √(4 - x)
Show Answer
Domain: 4 - x ≥ 0, so x ≤ 4
Domain: (-∞, 4]
Range: Since √(4-x) ≥ 0, and as x approaches -∞, the value increases
Range: [0, ∞)
Problem 2
Find the domain of f(x) = 1/(x² - 9)
Show Answer
Domain: x² - 9 ≠ 0, so x ≠ ±3
Domain: (-∞, -3) ∪ (-3, 3) ∪ (3, ∞)
Tips for Finding Domain and Range
- ✓Always check for restrictions: Division by zero, negative square roots, and invalid log arguments
- ✓Use interval notation: It's the clearest way to express domain and range
- ✓Graph when possible: Visualizing the function helps identify range
- ✓Consider function composition: When functions are combined, restrictions may compound
- ✓Test boundary values: Check what happens at endpoints and excluded values
Conclusion
Understanding domain and range is fundamental to working with functions in mathematics. Whether you're studying algebra, calculus, or advanced mathematics, being able to identify and express domain and range using interval notation is an essential skill.
Practice identifying restrictions, graphing functions, and expressing solutions clearly. With time, finding domain and range will become intuitive, and you'll be able to analyze even complex functions with confidence.