{"id":1654,"date":"2024-06-01T01:09:03","date_gmt":"2024-06-01T01:09:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1654"},"modified":"2025-08-13T23:16:52","modified_gmt":"2025-08-13T23:16:52","slug":"inverse-functions-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/inverse-functions-apply-it-1\/","title":{"raw":"Inverse Functions: Apply It 1","rendered":"Inverse Functions: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Confirm that two functions are inverses<\/li>\r\n \t<li>Figure out the allowed inputs and outputs for an inverse function and adjust the original function's domain to ensure it is one-to-one<\/li>\r\n \t<li>Discover or calculate the inverse of a function<\/li>\r\n \t<li>Draw the inverse of a function on the same graph by reflecting it across the line [latex]y=x[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>The Chinook salmon, a vital species in California's ecosystem, faced a critical threat of extinction. To protect these remarkable fish, conservationists launched an ambitious reintroduction program in California's Central Valley.[footnote]https:\/\/www.fisheries.noaa.gov\/feature-story\/endangered-winter-run-chinook-salmon-increase-millions-offspring-headed-sea[\/footnote]\r\n\r\n[caption id=\"attachment_6780\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-6780 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16235806\/6.3_Inverse_Functions_fishy-300x200.png\" alt=\"\" width=\"300\" height=\"200\" \/> Picture of a Chinook Salmon being held[\/caption]\r\n\r\nThe project centered on releasing juvenile Chinook salmon into a tributary of the Sacramento River. By monitoring the number of adult salmon returning to spawn each year, scientists could track the program's success and the species' recovery.\r\n<p class=\"whitespace-pre-wrap break-words\">Scientists collected data about the relationship between the number of juvenile salmon released and the number of adult salmon that returned to spawn. This relationship can be modeled using a function [latex]f[\/latex], where [latex]f(x)[\/latex] represents the number of returning adult salmon when [latex]x[\/latex] thousand juvenile salmon are released.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Using this model, biologists can better understand:<\/p>\r\n\r\n<ul class=\"-mt-1 [li&gt;&amp;]:mt-2 [li&gt;mark&gt;&amp;]:mt-2 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">How many juvenile salmon they need to release to achieve their target adult population<\/li>\r\n \t<li class=\"whitespace-normal break-words\">What factors might be affecting the survival rate<\/li>\r\n \t<li class=\"whitespace-normal break-words\">How to optimize their conservation efforts<\/li>\r\n<\/ul>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]31691[\/ohm2_question]<\/section><section aria-label=\"Try It\">\r\n<p class=\"whitespace-pre-wrap break-words\">The function [latex]f[\/latex] models how juvenile salmon releases ([latex]x[\/latex] thousand fish) relate to returning adult salmon ([latex]f(x)[\/latex]). However, biologists often need to work backward - if they want a specific number of adult salmon to return, they need to calculate how many juveniles to release.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">This is where inverse functions become valuable. The inverse function [latex]f^{-1}[\/latex] helps determine how many thousand juvenile salmon need to be released to achieve a target number of returning adults.<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]31692[\/ohm2_question]<\/section><section aria-label=\"Try It\">Now that we understand how to interpret specific values of our inverse function, let's examine its overall behavior. This will help biologists better understand the relationship between juvenile releases and adult returns across different population sizes.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]31693[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Confirm that two functions are inverses<\/li>\n<li>Figure out the allowed inputs and outputs for an inverse function and adjust the original function&#8217;s domain to ensure it is one-to-one<\/li>\n<li>Discover or calculate the inverse of a function<\/li>\n<li>Draw the inverse of a function on the same graph by reflecting it across the line [latex]y=x[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p>The Chinook salmon, a vital species in California&#8217;s ecosystem, faced a critical threat of extinction. To protect these remarkable fish, conservationists launched an ambitious reintroduction program in California&#8217;s Central Valley.<a class=\"footnote\" title=\"https:\/\/www.fisheries.noaa.gov\/feature-story\/endangered-winter-run-chinook-salmon-increase-millions-offspring-headed-sea\" id=\"return-footnote-1654-1\" href=\"#footnote-1654-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<figure id=\"attachment_6780\" aria-describedby=\"caption-attachment-6780\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6780 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16235806\/6.3_Inverse_Functions_fishy-300x200.png\" alt=\"\" width=\"300\" height=\"200\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16235806\/6.3_Inverse_Functions_fishy-300x200.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16235806\/6.3_Inverse_Functions_fishy-768x512.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16235806\/6.3_Inverse_Functions_fishy-65x43.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16235806\/6.3_Inverse_Functions_fishy-225x150.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16235806\/6.3_Inverse_Functions_fishy-350x233.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/16235806\/6.3_Inverse_Functions_fishy.png 800w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-6780\" class=\"wp-caption-text\">Picture of a Chinook Salmon being held<\/figcaption><\/figure>\n<p>The project centered on releasing juvenile Chinook salmon into a tributary of the Sacramento River. By monitoring the number of adult salmon returning to spawn each year, scientists could track the program&#8217;s success and the species&#8217; recovery.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Scientists collected data about the relationship between the number of juvenile salmon released and the number of adult salmon that returned to spawn. This relationship can be modeled using a function [latex]f[\/latex], where [latex]f(x)[\/latex] represents the number of returning adult salmon when [latex]x[\/latex] thousand juvenile salmon are released.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Using this model, biologists can better understand:<\/p>\n<ul class=\"-mt-1 [li&gt;&amp;]:mt-2 [li&gt;mark&gt;&amp;]:mt-2 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">How many juvenile salmon they need to release to achieve their target adult population<\/li>\n<li class=\"whitespace-normal break-words\">What factors might be affecting the survival rate<\/li>\n<li class=\"whitespace-normal break-words\">How to optimize their conservation efforts<\/li>\n<\/ul>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm31691\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=31691&theme=lumen&iframe_resize_id=ohm31691&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<p class=\"whitespace-pre-wrap break-words\">The function [latex]f[\/latex] models how juvenile salmon releases ([latex]x[\/latex] thousand fish) relate to returning adult salmon ([latex]f(x)[\/latex]). However, biologists often need to work backward &#8211; if they want a specific number of adult salmon to return, they need to calculate how many juveniles to release.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">This is where inverse functions become valuable. The inverse function [latex]f^{-1}[\/latex] helps determine how many thousand juvenile salmon need to be released to achieve a target number of returning adults.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm31692\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=31692&theme=lumen&iframe_resize_id=ohm31692&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">Now that we understand how to interpret specific values of our inverse function, let&#8217;s examine its overall behavior. This will help biologists better understand the relationship between juvenile releases and adult returns across different population sizes.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm31693\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=31693&theme=lumen&iframe_resize_id=ohm31693&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1654-1\">https:\/\/www.fisheries.noaa.gov\/feature-story\/endangered-winter-run-chinook-salmon-increase-millions-offspring-headed-sea <a href=\"#return-footnote-1654-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":12,"menu_order":23,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":142,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1654"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1654\/revisions"}],"predecessor-version":[{"id":7686,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1654\/revisions\/7686"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/142"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1654\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1654"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1654"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1654"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1654"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}