[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-605568-105":59,"doc-detail-605568-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","chapter-7-inverse-functions-one-to-one-functions","Chapter 7 - Inverse Functions - One-to-One Functions","","Chapter 7 introduces inverse functions as the mathematical “undo” of a function’s original mapping. It reviews what it means for a function to be a function using the vertical line test, then develops one-to-one functions using the horizontal line test and the requirement that each y-value correspond to exactly one x-value. Examples include the cube root behavior and restricting the domain for f(x)=x^2. The chapter then defines inverse notation and explains how inverse composition reverses inputs to recover the starting value.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/chapter-7-inverse-functions-one-to-one-functions/605568/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/chapter-7-inverse-functions-one-to-one-functions/605568.png","ImageObject",300,407,{"name":92,"@type":93},"RuangKosong","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-21","2026-10-16",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How does a vertical line test determine whether a mapping is a function?","Question",{"text":112,"@type":113},"A vertical line test checks whether an x-value corresponds to exactly one y-value. If a vertical line crosses the graph more than once, the relation fails the function definition.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is a one-to-one function, and how is it tested graphically?",{"text":117,"@type":113},"A function is one-to-one when every y-value in its range corresponds to exactly one x-value in its domain. Graphically, it passes the horizontal line test: a horizontal line should intersect the graph at most once.",{"name":119,"@type":110,"acceptedAnswer":120},"How can the function f(x)=x^2 be modified to become one-to-one?",{"text":121,"@type":113},"Restrict the domain so the graph is taken only on one side of the axis. For example, using x ≥ 0 or x ≤ 0 removes the symmetry that causes multiple x-values to share the same y-value.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},605568,1792581562,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":36},962090883219,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Chapter 7  \nInverse Functions  \nOne-to-One Functions  \nFinally, we get to the chapter that I’ve been wanting to teach for so long, and the concepts that got me started writing this book in the first place. Inverse functions essentially “undo” the work that the function did in the first place. They bring us back to square one, so to speak.  \nSince this book started with functions, and the inverse is going to undo the function, it would be good to review what a function is. A function takes an element from a particular domain, and sends it over to exactly one element in a particular range. That’s rough, but graphically we show a function very easily with the vertical line test. If a vertical line crosses the graph in more than one place, then we have an element from the domain going to two or more elements in the range and thus do not have a function. We can say this also in this way; if an x-value goes to two or more y-values, then we don’t have a function.  \nIn the above example, it may seem like the vertical line x = 0 is going through more than one point, just because the curved line appears to get so vertical at that point, but really, the graph is of the cube root function, and only one x-value has a cube root of 0, and that is when x is zero.  \nAgain, we have declared and defined a function to have every x-value go to exactly one y-value. Those two words “exactly one” are critical to the definition of a function.  \nBut if we want to go the other way, to undo a function, then it makes sense to have every y-value go to exactly one x-value, otherwise we might not have a function. We basically go the opposite way. We call this type of function a one-toone function:  \nDefinition: A function f is one-to-one if for every y-value in the range off, there is exactly one x-value in the domain off such that f (x) = y.  \nNotice that we started the definition with the words “A function…”. That means that we already pass the vertical line test, that we have a function.  \nNow, I’ve read math textbooks before, and I doubt that you reading mine is your first time around as well. And I know that by this time a picture is going to bea great help, rather than more words. So to give you a picture of what we mean here, I’m going to start by showing you a function that is NOT one-to-one:  \nThis is of course our favorite f (x) = x 2 function. Notice how f (−2) and f (2) both equal the same thing, which is 4? This means that this is NOT one-to-one; I have  \none y-value pairing up to two x-values. The y-value, the height, is the same for both x-values.  \nSince both f (−2) and f (2) are at the same height, at 4, then we can perhaps see that a horizontal line could be drawn from (−2,4) to (2,4), and indeed we can:  \nThis drawing of a horizontal line is then our test for one-to-one-ness; if we can draw such a line, then the function is not one-to-one. We call this the Horizontal Line Test, just as with our test for functions way back we called a similar test the Vertical Line Test.  \nYou will see in the next section that we must require one-to-one-ness in order to determine the inverse of a function. But, while we are here, is there a way to make this function one-to-one?  \nYes, there is. Our problem with f (x) = x 2 is that the function comes down, kisses the x-axis at (0,0) and then symmetrically swings back up again. If we get rid of the “coming down” part, and just start the function at (0,0), then we’ve got a one-to-one function:  \nWhat we had to do was chop off our domain, called restricting the domain, and essentially refuse to define the function whenever x is less than zero. But as you can see, now every horizontal line goes through the graph at exactly one point (that is, every horizontal line that is above the x-axis).  Now we have a one-to-one function.  \nMake up your own function, using some graphing software, and determine whether the function is one-to-one or not. If it isn’t, practice restricting the domain so that it becomes one-to-on","cbCaid4jVeHlzxs0","https://ap.wps.com/l/cbCaid4jVeHlzxs0","pdf",4553894,16,"English","# Chapter 7: Inverse Functions\n## One-to-One Functions\n## Definition of an Inverse Function","[{\"question\":\"How does a vertical line test determine whether a mapping is a function?\",\"answer\":\"A vertical line test checks whether an x-value corresponds to exactly one y-value. If a vertical line crosses the graph more than once, the relation fails the function definition.\"},{\"question\":\"What is a one-to-one function, and how is it tested graphically?\",\"answer\":\"A function is one-to-one when every y-value in its range corresponds to exactly one x-value in its domain. Graphically, it passes the horizontal line test: a horizontal line should intersect the graph at most once.\"},{\"question\":\"How can the function f(x)=x^2 be modified to become one-to-one?\",\"answer\":\"Restrict the domain so the graph is taken only on one side of the axis. For example, using x ≥ 0 or x ≤ 0 removes the symmetry that causes multiple x-values to share the same y-value.\"}]","Chapter 7 - Inverse Functions - One-to-One Functions | PDF",1792132724]