{"title":"Composition of Functions","authors":"Tom Kelliher","doi":"10.1201/9781315273761-23","DOIUrl":"https://doi.org/10.1201/9781315273761-23","url":null,"abstract":"1. Present the following explanation to students: Consider the function x x h 4 ) ( . Using an input of 9 to evaluate h, we see that . 6 36 ) 9 ( 4 ) 9 ( h So, h(9) = 6. Since we performed two operations—multiplication and finding the square root—we can think of h as a composite of two functions. Let’s call these two functions f(x) and g(x), with f(x) = 4x and g(x) = x . We can evaluate h(9) by finding the output of f(9) and using that output as the input of function g. First, use 9 as the input for f, and find f(9) = 4(9) = 36. Next, use that output as the input for g and find 6 36 ) 36 ( g . Therefore, 6 ) 36 ( )] 9 ( [ ) 9 ( ) 9 ( g f g f g h . When we use an output of one function as an input for another function, we are creating a composition of functions. In our example, h is a composition of f and g, which is written as ) ( ) ( or )] ( [ ) ( x f g x h x f g x h . Both equations are read “h(x) equals g of f of x.”","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126044880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Denumerable and Countable Sets","authors":"","doi":"10.1201/9781315273761-34","DOIUrl":"https://doi.org/10.1201/9781315273761-34","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124049184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Real Number System","authors":"K. E. Hummel","doi":"10.1201/9781315273761-31","DOIUrl":"https://doi.org/10.1201/9781315273761-31","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"151 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132799243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quantifiers and Predicates Chapter","authors":"","doi":"10.1201/9781315273761-9","DOIUrl":"https://doi.org/10.1201/9781315273761-9","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133130760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Systems of Whole and Natural Numbers","authors":"","doi":"10.1201/9781315273761-27","DOIUrl":"https://doi.org/10.1201/9781315273761-27","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133833193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Logic and Propositional Calculus","authors":"","doi":"10.1201/9781315273761-7","DOIUrl":"https://doi.org/10.1201/9781315273761-7","url":null,"abstract":"Propositional logic was eventually refined using symbolic logic. The 17th/18th century philosopher Gottfried Leibniz (an inventor of calculus) has been credited with being the founder of symbolic logic. Although his work was the first of its kind, it was unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were re-achieved by logicians like George Boole and Augustus De Morgan in the 19th century completely independent of Leibniz.","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117060803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Image and Preimage Functions","authors":"","doi":"10.1201/9781315273761-24","DOIUrl":"https://doi.org/10.1201/9781315273761-24","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"73 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114092146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalized Union and Intersection","authors":"","doi":"10.1201/9781315273761-17","DOIUrl":"https://doi.org/10.1201/9781315273761-17","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133281846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Least Upper Bound and Greatest Lower Bound","authors":"E. G. Powerset","doi":"10.1201/9781315273761-39","DOIUrl":"https://doi.org/10.1201/9781315273761-39","url":null,"abstract":"","PeriodicalId":348406,"journal":{"name":"Introductory Concepts for Abstract Mathematics","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130459212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
求助内容：
应助结果提醒方式：