Functions5.1Relations andFunctionsIn this section, you will learn about the relationship between relations and functions. Upon completion you will beable to:•State whether or not a relation is a function.•Use and apply function notation to given scenarios.•Translate between a set of real numbers and interval notation.•State the domain and range of a given graphical representation of a function, using interval notation.WritingIntervalNotationThis textbook focuses on the use of the real numbers,R. Recall that we may visualizeRas a line. Segments of thisline are calledintervalsof numbers. Below is a summary of theinterval notationassociated with given sets ofnumbers.For intervals with finite endpoints, we list the left endpoint, then the right endpoint. We use square brackets, ‘[’ or‘]’, if the endpoint is included in the interval and use a filled-in or ‘closed’ dot to indicate membership in the interval.Otherwise, we use parentheses, ‘(’ or ‘)’ and an ‘open’ circle to indicate that the endpoint is not part of the set.If the interval does not have finite endpoints, we use the symbol-∞to indicate that the interval extends indefinitelyto the left and∞to indicate that the interval extends indefinitely to the right. Since infinity is a concept, and not anumber, we always use parentheses when using these symbols in interval notation, and use an appropriate arrow toindicate that the interval extends indefinitely in one (or both) directions.Set-BuilderSegment of theIntervalVerbalNotationReal Number LineNotationDescription{x|a<x<b}ab◦◦(a,b)all numbers betweenaandb{x|a≤x<b}ab•◦[a,b)all numbers betweenaandb, includinga{x|a<x≤b}ab◦•(a,b]all numbers betweenaandb, includingb{x|a≤x≤b}ab••[a,b]all numbers betweenaandb, includingaandb{x|x<b}b◦(-∞,b)all numbers strictly less thanb{x|x≤b}b•(-∞,b]all numbers less than or equal tob{x|x>a}a◦(a,∞)all numbers strictly greater thana{x|x≥a}a•[a,∞)all numbers greater than or equal toa{x| -∞<x<∞}(-∞,∞)all real number©TAMU98

5.1 Relations and FunctionsDefinition•Set-builder notationis a method of specifying a set of elements that satisfy a certain condition. It takesthe form{x|statement aboutx}which is read as, “ the set of allxsuch that the statement aboutxis true".For example,{x|4<x≤12}would be read as “the set of allxsuch that four is less than x is less than or equal to 12".•Interval notationis a way of describing sets that include all real numbers between a lower limit that mayor may not be included and an upper limit that may or may not be included. The endpoint values are listedbetween brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicatesexclusion from the set. The example, given in set-builder notation above, would be written using intervalnotation as follows,(4,12]

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