A closed interval is a set of real numbers that includes both the lower and upper bounds of the interval. It is denoted by square brackets, [a, b], where 'a' represents the lower bound and 'b' represents the upper bound of the interval. The closed interval includes all the real numbers between 'a' and 'b', as well as the endpoints 'a' and 'b' themselves.
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Closed intervals are important in the context of domain and range because they represent the set of all real numbers that satisfy a given condition or constraint.
The domain of a function is the set of all input values for which the function is defined, and it can be represented as a closed interval.
The range of a function is the set of all possible output values, and it can also be represented as a closed interval.
Closed intervals are often used to describe the boundaries or limits of a function's behavior, such as the minimum and maximum values it can take.
Closed intervals are essential in the study of functions, as they help in understanding the behavior of the function within a specific range of input values.
Review Questions
Explain how closed intervals are used to represent the domain of a function.
The domain of a function is the set of all input values for which the function is defined. Closed intervals are often used to represent the domain of a function because they include the lower and upper bounds of the interval, which are the minimum and maximum values that the input can take. For example, the domain of the function $f(x) = \sqrt{x}$ can be represented as the closed interval [0, \infty), indicating that the input values can be any non-negative real number, including 0.
Describe the relationship between closed intervals and the range of a function.
The range of a function is the set of all possible output values that the function can produce. Similar to the domain, the range of a function can often be represented as a closed interval. This is because the range is bounded by the minimum and maximum values that the function can take, and a closed interval includes these endpoints. For example, the range of the function $f(x) = x^2$ on the domain [-2, 2] can be represented as the closed interval [0, 4], as the function values are always non-negative and lie between 0 and 4, inclusive.
Analyze how the use of closed intervals can help in understanding the behavior of a function.
Closed intervals are crucial in understanding the behavior of a function because they provide information about the limits or boundaries of the function's values. By identifying the closed interval that represents the domain or range of a function, you can determine the minimum and maximum values the function can take, as well as any constraints or restrictions on the input or output values. This understanding of the function's behavior within a specific closed interval can help in analyzing its properties, such as its monotonicity, extrema, and graphical representation, which are essential in solving problems and making inferences about the function's behavior.
An open interval is a set of real numbers that excludes the lower and upper bounds of the interval. It is denoted by parentheses, (a, b), where 'a' represents the lower bound and 'b' represents the upper bound of the interval. The open interval includes all the real numbers between 'a' and 'b', but not the endpoints 'a' and 'b'.
Half-Open Interval: A half-open interval is a set of real numbers that includes one of the endpoints and excludes the other. It is denoted by a square bracket and a parenthesis, [a, b) or (a, b], where 'a' represents the lower bound and 'b' represents the upper bound of the interval.
Interval notation is a way of representing intervals using mathematical symbols, such as square brackets, parentheses, and commas, to indicate the inclusion or exclusion of the endpoints.