Fundamentals of Mathematics in Medical Research: Theory and Cases

Functions using a Single Variable

Author(s): Carlos Polanco * .

Pp: 1-9 (9)

DOI: 10.2174/9789815223132124010005

* (Excluding Mailing and Handling)

Abstract

In this chapter, we will delve into a variety of fundamental subjects related to the study of real functions and the transformations of such functions. To begin, we will investigate the absolute value, as well as the absolute value function, and examine the behavior and attributes of each of these concepts. Next, we will delve deeper into the world of functions, concentrating on the functions that are associated with a single variable and the graphical representations of these functions. As we move further, we will investigate the most important real-valued functions, such as polynomial functions and piecewise-defined functions, amongst others. Because of these functions, we are able to acquire a deeper comprehension of transformations and how they have an effect on function graphs. In this lesson, we will investigate the scaling and translating transformations, focusing on how they affect the features of functions. Examples and activities are presented throughout the chapter to help readers better understand the fundamental ideas behind graphical representation and function analysis.

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      • 13. app/Models/UserAccess/UserAccess.php:1266
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      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
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      580μsalphaeurekaselec_live_10_06_2022UserAccess.php#1266
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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 3741 and `b`.`banner_from_date` <= 1740514496 and `b`.`banner_to_date` >= 1740514496 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.05msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 3741
      • 3: 1740514496
      • 4: 1740514496
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='e9e4444473c2e5a6b125869a3693b647'
      180μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        Ta20VxMP2UG6OsM93pAqBTgrZnpnmr1tpBbvH0CU
        uc_cart_id
        e9e4444473c2e5a6b125869a3693b647
        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/22834" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /chapter/22834
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:17 [ "x-https" => array:1 [ 0 => "1" ] "host" => array:1 [ 0 => "alpha.eurekaselect.com" ] "priority" => array:1 [ 0 => "u=0, i" ] "cookie" => array:1 [ 0 => "wayfindertarget=eyJpdiI6IjRzaHUzdEVMRWwyV3dDWjJpeWQ4aUE9PSIsInZhbHVlIjoiZjViNGY5dzRBY1BDRXNocWlNeGlHWFZOeDdTeWgxcjNhRkF5STFockFKTT0iLCJtYWMiOiIwMDU3MmI5NTQ5Mjg2MjE4MjBjMjhiZDE2ZGE0ZDU2OGZjOGJiNDE5NjkzM2M1OWI0ZDQzODBhNWVjZjVjODJlIiwidGFnIjoiIn0%3Dwayfindertarget=eyJpdiI6IjRzaHUzdEVMRWwyV3dDWjJpeWQ4aUE9PSIsInZhbHVlIjoiZjViNGY5dzRBY1BDRXNocWlNeGlHWFZOeDdTeWgxcjNhRkF5STFockFKTT0iLCJtYWMiOiIwMDU3MmI5NTQ5Mjg2" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] ]
        request_cookies
        0 of 0
        array:1 [ "wayfindertarget" => null ]
        response_headers
        0 of 0
        array:5 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "no-cache, private" ] "date" => array:1 [ 0 => "Tue, 25 Feb 2025 20:14:56 GMT" ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6ImIzZ1hSY0lCeHVGajdjZE5zdHRkT3c9PSIsInZhbHVlIjoiTk44TDVVUVg4ckxvZkMxbmlVNlp5aXB1RUxnVzBHOTJESzdBTlNqbUJaeDFnOGlORWlaZ2l2T2IwTUtDMVZWSWdveDkyRzNHODMyUklQM1lPMUtXejhYUlg3YVRTRE1SSXdQMng1M1dXSEw1T3Y5M0tYdWkyL1ppcHNsaURpU0siLCJtYWMiOiIxNjk0NDQyN2UyMTNjZDg0OTkxOGNlYzY0MTUzZmMwNDE4MTljNjBiZDA0ZjU4NWYyYTQyMzEyNjY3NTVlMDM5IiwidGFnIjoiIn0%3D; expires=Fri, 28 Feb 2025 08:14:56 GMT; Max-Age=216000; path=/; secure; samesite=laxXSRF-TOKEN=eyJpdiI6ImIzZ1hSY0lCeHVGajdjZE5zdHRkT3c9PSIsInZhbHVlIjoiTk44TDVVUVg4ckxvZkMxbmlVNlp5aXB1RUxnVzBHOTJESzdBTlNqbUJaeDFnOGlORWlaZ2l2T2IwTUtDMVZWSWdveDkyR" 1 => "alphaeurekaselectnet_session=eyJpdiI6IkNVaXhQcmMzekFVcjJkajVadHBvTXc9PSIsInZhbHVlIjoiQWJ0d21wbElVTTdxWmNvNWFLZGYvL0U5aU56NXVaOWtLRUpxMGhYY0dldGNnTjdXdDI1NXJockxTK0ZwT0xSaTRQQnl4TXdiSW1VV3lEMGQvRENOTGZBdURaWHpFbnJLR0M3dm5iVGp5amliZ1YzZjF5NGUrRVR1QUZBeTU3b3giLCJtYWMiOiJjODk0YmVjYzYwNGM2MDA3OTE0NjM0YzIyZWZjZjZlYmRlODBkM2JhMTE3ZWU1NTMwOTJkMjhlMjVhMmU5MDBkIiwidGFnIjoiIn0%3D; path=/; secure; httponly; samesite=laxalphaeurekaselectnet_session=eyJpdiI6IkNVaXhQcmMzekFVcjJkajVadHBvTXc9PSIsInZhbHVlIjoiQWJ0d21wbElVTTdxWmNvNWFLZGYvL0U5aU56NXVaOWtLRUpxMGhYY0dldGNnTjdXdDI1NXJockx" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6ImIzZ1hSY0lCeHVGajdjZE5zdHRkT3c9PSIsInZhbHVlIjoiTk44TDVVUVg4ckxvZkMxbmlVNlp5aXB1RUxnVzBHOTJESzdBTlNqbUJaeDFnOGlORWlaZ2l2T2IwTUtDMVZWSWdveDkyRzNHODMyUklQM1lPMUtXejhYUlg3YVRTRE1SSXdQMng1M1dXSEw1T3Y5M0tYdWkyL1ppcHNsaURpU0siLCJtYWMiOiIxNjk0NDQyN2UyMTNjZDg0OTkxOGNlYzY0MTUzZmMwNDE4MTljNjBiZDA0ZjU4NWYyYTQyMzEyNjY3NTVlMDM5IiwidGFnIjoiIn0%3D; expires=Fri, 28-Feb-2025 08:14:56 GMT; path=/; secureXSRF-TOKEN=eyJpdiI6ImIzZ1hSY0lCeHVGajdjZE5zdHRkT3c9PSIsInZhbHVlIjoiTk44TDVVUVg4ckxvZkMxbmlVNlp5aXB1RUxnVzBHOTJESzdBTlNqbUJaeDFnOGlORWlaZ2l2T2IwTUtDMVZWSWdveDkyR" 1 => "alphaeurekaselectnet_session=eyJpdiI6IkNVaXhQcmMzekFVcjJkajVadHBvTXc9PSIsInZhbHVlIjoiQWJ0d21wbElVTTdxWmNvNWFLZGYvL0U5aU56NXVaOWtLRUpxMGhYY0dldGNnTjdXdDI1NXJockxTK0ZwT0xSaTRQQnl4TXdiSW1VV3lEMGQvRENOTGZBdURaWHpFbnJLR0M3dm5iVGp5amliZ1YzZjF5NGUrRVR1QUZBeTU3b3giLCJtYWMiOiJjODk0YmVjYzYwNGM2MDA3OTE0NjM0YzIyZWZjZjZlYmRlODBkM2JhMTE3ZWU1NTMwOTJkMjhlMjVhMmU5MDBkIiwidGFnIjoiIn0%3D; path=/; secure; httponlyalphaeurekaselectnet_session=eyJpdiI6IkNVaXhQcmMzekFVcjJkajVadHBvTXc9PSIsInZhbHVlIjoiQWJ0d21wbElVTTdxWmNvNWFLZGYvL0U5aU56NXVaOWtLRUpxMGhYY0dldGNnTjdXdDI1NXJockx" ] ]
        session_attributes
        0 of 0
        array:5 [ "_token" => "Ta20VxMP2UG6OsM93pAqBTgrZnpnmr1tpBbvH0CU" "uc_cart_id" => "e9e4444473c2e5a6b125869a3693b647" "_previous" => array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/22834" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]
        ClearShow all
        Date ↕MethodURLData
        #12025-02-25 20:14:56GET/chapter/22834112182