Interval notation is a way of representing a set of real numbers that fall within a specific range or interval on the number line. It is a concise and precise method of describing the domain or range of a function.
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Interval notation uses square brackets [ ] to represent closed intervals, where the endpoints are included, and parentheses ( ) to represent open intervals, where the endpoints are excluded.
Interval notation can be used to describe both bounded and unbounded intervals on the number line.
The order of the endpoints in interval notation is important, with the smaller value coming before the larger value.
Interval notation can be used to describe the domain and range of a function, which are essential for understanding the function's behavior and limitations.
Understanding interval notation is crucial for solving problems involving inequalities, graphing functions, and analyzing the behavior of mathematical models.
Review Questions
How does interval notation represent closed and open intervals on the number line?
In interval notation, closed intervals are represented using square brackets [ ], where the endpoints are included in the set. For example, [3, 7] represents the set of all real numbers from 3 to 7, including 3 and 7. Open intervals are represented using parentheses ( ), where the endpoints are excluded from the set. For example, (2, 5) represents the set of all real numbers greater than 2 and less than 5, but not including 2 or 5.
Explain how interval notation can be used to describe the domain and range of a function.
The domain of a function is the set of all possible input values, and the range is the set of all possible output values. Interval notation can be used to concisely and precisely describe these sets. For example, if a function $f(x)$ is defined for all real numbers greater than or equal to -2 and less than or equal to 5, the domain can be expressed as the closed interval $[-2, 5]$. Similarly, if the function $g(x)$ has a range of all real numbers greater than 3 and less than 10, the range can be expressed as the open interval $(3, 10)$.
Analyze how the order of the endpoints in interval notation affects the interpretation of the set of real numbers represented.
The order of the endpoints in interval notation is crucial, as it determines the direction and orientation of the interval on the number line. If the smaller value comes before the larger value, the interval represents the set of real numbers from the smaller value to the larger value. However, if the larger value comes before the smaller value, the interval represents the empty set, as there are no real numbers that satisfy the condition. This understanding of the order of endpoints is essential for correctly interpreting and working with interval notation, particularly when solving problems involving inequalities and analyzing the behavior of functions.