Open Interval

Definition

An open interval is a set of real numbers that includes all the values between two specified endpoints, but does not include the endpoints themselves. It is denoted using parentheses to indicate the exclusion of the endpoints.

5 Must Know Facts For Your Next Test

1. Open intervals are commonly represented using the notation $(a, b)$, where $a$ and $b$ are the endpoints, and $a < b.
2. The set of real numbers in an open interval $(a, b)$ is the set of all $x$ such that $a < x < b.
3. Open intervals are useful in the context of solving rational inequalities, as they help define the regions where the inequality is true.
4. When solving rational inequalities, the solutions are often expressed as a union of open intervals.
5. Open intervals are important in the study of function behavior, as they can be used to determine the domain and range of a function.

Review Questions

• Explain how open intervals are used in the context of solving rational inequalities.
  • When solving rational inequalities, the solutions are often expressed as a union of open intervals. This is because open intervals exclude the endpoints, which allows for the representation of the regions where the inequality is true. For example, the solution to the inequality $\frac{x - 2}{x + 3} > 0$ would be expressed as the open interval $(2, -3)$, indicating that the inequality is true for all values of $x$ between 2 and -3, but not including the endpoints.
• Describe the differences between open intervals and closed intervals, and how they are represented.
  • The key difference between open intervals and closed intervals is the inclusion or exclusion of the endpoints. Open intervals, denoted using parentheses $(a, b)$, include all the values between the two endpoints $a$ and $b$, but do not include the endpoints themselves. Closed intervals, denoted using square brackets $[a, b]$, include all the values between the two endpoints, including the endpoints. This distinction is important when working with functions and inequalities, as open intervals can be used to define the domain and range of a function, or the regions where an inequality is true.
• Analyze the role of open intervals in the study of function behavior, particularly in the context of determining the domain and range of a function.
  • Open intervals play a crucial role in the study of function behavior, as they can be used to define the domain and range of a function. The domain of a function is the set of input values for which the function is defined, and this is often expressed as a union of open intervals. Similarly, the range of a function is the set of output values that the function can produce, and this can also be represented using open intervals. Understanding the properties of open intervals, such as their exclusion of the endpoints, is essential for analyzing the behavior of functions and determining their key characteristics.

Related terms

Closed Interval: A closed interval is a set of real numbers that includes all the values between two specified endpoints, including the endpoints themselves. It is denoted using square brackets to indicate the inclusion of the endpoints.

Bounded Interval: A bounded interval is an interval that has a finite length, with both a lower and upper bound.

Unbounded Interval: An unbounded interval is an interval that has no finite upper or lower bound, extending infinitely in one or both directions.