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Write Domain and Range in Interval Notation Calculator

Reviewed by Calculator Editorial Team

Interval notation is a concise way to represent sets of real numbers. This guide explains how to properly write the domain and range of functions using interval notation, with practical examples and a built-in calculator.

What is Interval Notation?

Interval notation is a mathematical shorthand for describing ranges of real numbers. It's commonly used in calculus, algebra, and other branches of mathematics to specify the domain and range of functions.

There are several types of interval notation:

  • Closed interval: [a, b] - includes all numbers from a to b, including a and b
  • Open interval: (a, b) - includes all numbers from a to b, excluding a and b
  • Half-open intervals: [a, b) and (a, b] - include one endpoint but not the other
  • Infinite intervals: [a, ∞) and (-∞, b] - include all numbers from a to infinity or negative infinity to b

Interval notation is different from set notation, which uses curly braces { } to list elements. For example, [1, 5] in interval notation represents the same set as {1, 2, 3, 4, 5} in set notation.

How to Write Domain and Range in Interval Notation

When writing the domain and range of a function, you need to consider:

  1. The values of x that are allowed as inputs (domain)
  2. The resulting values of y that the function produces (range)

Steps to Write Domain and Range

  1. Identify the function's restrictions (if any) to determine the domain
  2. Use interval notation to express the domain
  3. Determine the minimum and maximum output values of the function
  4. Use interval notation to express the range
Domain: [a, b] or (a, b) or other appropriate interval Range: [c, d] or (c, d) or other appropriate interval

Common Domain and Range Examples

For a linear function f(x) = 2x + 3:

  • Domain: All real numbers (-∞, ∞)
  • Range: All real numbers (-∞, ∞)

For a quadratic function f(x) = x²:

  • Domain: All real numbers (-∞, ∞)
  • Range: [0, ∞) because the minimum value is 0

Examples

Example 1: Linear Function

Function: f(x) = 3x - 2

  • Domain: All real numbers (-∞, ∞)
  • Range: All real numbers (-∞, ∞)

Example 2: Square Root Function

Function: f(x) = √(x - 1)

  • Domain: [1, ∞) because the expression under the square root must be non-negative
  • Range: [0, ∞) because the square root function outputs non-negative values

Example 3: Rational Function

Function: f(x) = 1/(x - 2)

  • Domain: All real numbers except x = 2 (-∞, 2) ∪ (2, ∞)
  • Range: All real numbers except y = 0 (-∞, 0) ∪ (0, ∞)

Common Mistakes

When writing domain and range in interval notation, avoid these common errors:

  1. Using incorrect interval brackets (open vs. closed)
  2. Forgetting to exclude values that make the function undefined
  3. Including values that are outside the actual range of the function
  4. Using set notation instead of interval notation

Always double-check your work by testing values at the endpoints of your intervals to ensure they satisfy the function's conditions.

FAQ

What is the difference between domain and range?
The domain refers to all possible input values (x-values) for a function, while the range refers to all possible output values (y-values) that the function can produce.
How do I know if an interval should be open or closed?
An interval should be closed (using square brackets) if the endpoint value is included in the domain or range. It should be open (using parentheses) if the endpoint value is excluded.
Can the domain or range be infinite?
Yes, both the domain and range can be infinite. For example, the domain of f(x) = x² is all real numbers (-∞, ∞), and the range is [0, ∞).