At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).Identify the intervals to be included in the set by determining where the heavy line overlays the real line.Given a line graph, describe the set of values using interval notation. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. The endpoint values are listed between brackets or parentheses. Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. However, in set notation, rather than using the symbol "∪," we use the word "or" by convention.\) which is read as, “the set of all x such that the statement about x is true.” For example, Like interval notation, we can also use unions in set builder notation. Which can be read as "the set of all y such that y is greater than or equal to zero." The range of f(x) = x 2 in set notation is: The above can be read as "the set of all x such that x is an element of the set of all real numbers." In other words, the domain is all real numbers. Using the same example as above, the domain of f(x) = x 2 in set notation is: Standard inequality symbols such as, ≥, and so on are also used in set notation. Indicates that an element is a member of some set "such that" - symbol is followed by a constraint Like interval notation, there are a number of symbols used in set notation, the most common of which are shown in the table below: When using set notation, also referred to as set builder notation, we use inequality symbols to describe the domain and range as a set of values. Note that it is also possible to use multiple union symbols to combine more intervals in the same manner. The domain of the function is therefore all x-values except those in the interval (0, 1), which we can indicate in interval notation using the union symbol as follows: This is the same as our function above, except that it is not defined over the interval (0, 1). In the context of interval notation, it simply means to combine two given intervals. The union symbol can be read as "or" and it is used throughout various fields of mathematics. The union symbol is used when we have a function whose domain or range cannot be described with just a single interval. The range can therefore be written in interval notation as: Recall that the range of f(x) = x 2 is all positive y-values, including 0. We used parentheses rather than brackets around each endpoint because the endpoints are negative and positive infinity, which by definition have no bound. In other words, any value from negative infinity to positive infinity will yield a real result. Recall that the domain of f(x) = x 2 is all real numbers. Let's look at the same example as above, f(x) = x 2 to see how interval notation is used. The endpoints are written between either parentheses or brackets, depending on whether the endpoint is included or not. ![]() The first term is the left endpoint and the second term is the right endpoint.The smallest term in the interval is written first, followed by a comma, and then the largest term.When indicating the domain in interval notation, we need to keep the following in mind: The table below shows the basic symbols used in interval notation and what they mean: When using interval notation, domain and range are written as intervals of values. Two of these notations are interval notation and set notation. ![]() This makes it far easier to express the domains and ranges of multiple functions at a time, particularly as functions get more complicated. ![]() While this is possible for all functions, different notations have been developed for expressing domains and ranges in a more concise way. Notice in the examples above that we described the domain and range using words. Thus, the range of f(x) = x 2 is all positive y-values. Then, from looking at the graph or testing a few x-values, we can see that any x-value we plug in will result in a positive y-value. Thus, the domain of f(x) = x 2 is all x-values. There are no x-values that will result in the function being undefined and matter what real x-value we plug in, the result will always be a real y-value.
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