Interval Notation

Definition

Interval notation is a mathematical notation used to represent a range of values or an interval on the number line. It uses parentheses and brackets to indicate whether endpoints are included or excluded from the interval, making it a concise way to express inequalities and piecewise functions. This notation is especially useful in contexts where functions are defined piecewise over different intervals of their domain.

5 Must Know Facts For Your Next Test

1. In interval notation, open intervals are represented with parentheses, such as (a, b), indicating that the endpoints 'a' and 'b' are not included in the interval.
2. Closed intervals use brackets, like [a, b], meaning that both 'a' and 'b' are included in the interval.
3. Half-open intervals combine both types, such as [a, b) or (a, b], which include one endpoint and exclude the other.
4. Interval notation can be used to describe domains for piecewise functions, specifying where each piece of the function is valid.
5. When working with piecewise functions, understanding how to convert between standard inequality form and interval notation is crucial for accurately describing function behavior.

Review Questions

• How does interval notation provide a clear representation of the domain for piecewise functions?
  • Interval notation provides a clear and concise way to express the domain of piecewise functions by specifying the ranges of input values associated with each piece of the function. For example, if a function has one expression defined for x < 0 and another for x ≥ 0, we can use interval notation to represent these domains as (-∞, 0) for the first piece and [0, ∞) for the second. This helps in quickly identifying where each part of the function applies without ambiguity.
• Explain how to translate a set of inequalities into interval notation when defining the domain of a piecewise function.
  • To translate a set of inequalities into interval notation, start by identifying each inequality's endpoint and whether they are included or excluded based on their type. For instance, if you have inequalities like x < 2 and x ≥ 3, you can represent this in interval notation as (-∞, 2) for the first part and [3, ∞) for the second part. When dealing with multiple pieces in a piecewise function, it's important to ensure that all intervals are non-overlapping and correctly reflect the conditions given by the inequalities.
• Evaluate how improper use of interval notation could lead to misinterpretations when defining piecewise functions.
  • Improper use of interval notation can significantly affect how piecewise functions are understood and graphed. For example, using (2, 5] instead of [2, 5] may imply that 2 is not included in the domain when it actually should be. Such errors can lead to misunderstandings about where a function is defined or what values it can take. Ensuring that interval notation accurately reflects whether endpoints are included or excluded is critical for conveying correct information about the function's behavior.

Related terms

Inequality: A mathematical statement that compares two expressions, often using symbols like <, >, ≤, or ≥ to show the relationship between them.

Domain: The set of all possible input values (or x-values) for which a function is defined.

Piecewise Function: A function that is defined by different expressions depending on the input value, often requiring the use of interval notation to specify each piece.