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Номер 8.7, страница 57 - гдз по алгебре 11 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
Популярные ГДЗ в 11 классе
Глава II. Степени и корни. Степенная функция. Параграф 8. Степенная функция, её свойства и график - номер 8.7, страница 57.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1124978 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/08-7" "field_display_title" => "8.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1124770 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Степени и корни. Степенная функция" "field_branch_order" => "II" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "39" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1058 #items: array:1 [ 0 => App\Models\Book {#1057 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4297 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1062 #items: array:1 [ 0 => App\Models\Term {#1059 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Term {#1071 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:2 [ 0 => App\Models\Term {#1076 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1077 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1069 #items: array:1 [ 0 => App\Models\Term {#1063 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1067 #items: array:1 [ 0 => App\Models\Term {#1075 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1078 #items: array:1 [ 0 => App\Models\Term {#1068 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Общественно-гуманитарного направления" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "530" "field_priority" => "2" "field_default_folder" => "/algebra_11/abylkasymova1-u19/" "field_isbn" => null "field_cover" => array:1 [ 0 => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 382 "subject" => 542 "class_subject" => 388 ] ] #original: array:50 [ "id" => 4297 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1062} "field_class" => Illuminate\Database\Eloquent\Collection {#1064} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1072} "field_author" => Illuminate\Database\Eloquent\Collection {#1065} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1073} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1069} "field_country" => Illuminate\Database\Eloquent\Collection {#1067} "field_city" => Illuminate\Database\Eloquent\Collection {#1078} "field_series" => Illuminate\Database\Eloquent\Collection {#1066} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1079} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1080} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1081} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1082} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Общественно-гуманитарного направления" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "530" "field_priority" => "2" "field_default_folder" => "/algebra_11/abylkasymova1-u19/" "field_isbn" => null "field_cover" => array:1 [ 0 => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1083} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1084} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1085} "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1086} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 382 "subject" => 542 "class_subject" => 388 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1048 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1124770 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Степени и корни. Степенная функция" "field_branch_order" => "II" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "39" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1058} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1048} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:2 [ 0 => App\Models\Branch {#1116 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1124770 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Степени и корни. Степенная функция" "field_branch_order" => "II" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "39" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1124770 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Степени и корни. Степенная функция" "field_branch_order" => "II" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1117} "field_page_start" => "39" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1118} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1120 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1124774 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Степенная функция, её свойства и график" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Term {#1121 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "53" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1124774 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Степенная функция, её свойства и график" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1124} "field_page_start" => "53" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1123} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1122} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:2 [ 0 => App\Models\Element {#1099 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1308185 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Edition {#1098 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:1 [ 0 => App\Models\Term {#1097 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1093 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1096} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1095} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1093} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1094} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1124978" "type" => "task" ] "text" => "<p><strong>8.7. 1)</strong> $f(x) = \frac{2}{(x-1)^2} + 4;$</p><p><strong>2)</strong> $f(x) = -\frac{3}{(x+2)^3} + 1,5;$</p><p><strong>3)</strong> $f(x) = (x+1)^{-\frac{3}{2}} - 2,5;$</p><p><strong>4)</strong> $f(x) = (x+1)^{-\frac{3}{4}} + 3,5.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "1401" "height" => 277 "path" => "/media/algebra_11/abylkasymova1-u19/0-08/7-1.webp?ts=1753819225" ] ] ] #original: array:7 [ "id" => 1308185 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089} "task" => array:2 [ "refs" => "1124978" "type" => "task" ] "text" => "<p><strong>8.7. 1)</strong> $f(x) = \frac{2}{(x-1)^2} + 4;$</p><p><strong>2)</strong> $f(x) = -\frac{3}{(x+2)^3} + 1,5;$</p><p><strong>3)</strong> $f(x) = (x+1)^{-\frac{3}{2}} - 2,5;$</p><p><strong>4)</strong> $f(x) = (x+1)^{-\frac{3}{4}} + 3,5.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "1401" "height" => 277 "path" => "/media/algebra_11/abylkasymova1-u19/0-08/7-1.webp?ts=1753819225" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1091 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1553194 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Edition {#1092 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5663 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 #items: array:1 [ 0 => App\Models\Term {#1090 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5663 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1055} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1124978" "type" => "task" ] "text" => "<p><strong>1)</strong> Для нахождения области определения функции $f(x) = \frac{2}{(x-1)^2} + 4$ необходимо учесть, что знаменатель дроби не может быть равен нулю. В данном случае знаменатель — это $(x-1)^2$. Приравняем его к нулю, чтобы найти недопустимые значения $x$:</p><p>$(x-1)^2 = 0$</p><p>$x-1 = 0$</p><p>$x = 1$</p><p>Таким образом, функция определена для всех действительных чисел $x$, кроме $x=1$.</p><p><strong>Ответ:</strong> $D(f) = (-\infty; 1) \cup (1; +\infty)$.</p><p><strong>2)</strong> Для функции $f(x) = -\frac{3}{(x+2)^3} + 1,5$ область определения также находится из условия, что знаменатель не должен обращаться в нуль. Знаменатель здесь — $(x+2)^3$. Найдем значение $x$, при котором он равен нулю:</p><p>$(x+2)^3 = 0$</p><p>$x+2 = 0$</p><p>$x = -2$</p><p>Следовательно, область определения функции — это множество всех действительных чисел, за исключением $x=-2$.</p><p><strong>Ответ:</strong> $D(f) = (-\infty; -2) \cup (-2; +\infty)$.</p><p><strong>3)</strong> Преобразуем функцию $f(x) = (x+1)^{-\frac{2}{3}} - 2,5$, используя свойство степени с отрицательным показателем $a^{-n} = \frac{1}{a^n}$ и свойство степени с рациональным показателем $a^{\frac{m}{n}} = \sqrt[n]{a^m}$:</p><p>$f(x) = \frac{1}{(x+1)^{\frac{2}{3}}} - 2,5 = \frac{1}{\sqrt[3]{(x+1)^2}} - 2,5$</p><p>Корень нечетной степени (здесь — кубический) определен для любого значения подкоренного выражения. Однако, так как это выражение находится в знаменателе, оно не должно быть равно нулю.</p><p>$\sqrt[3]{(x+1)^2} = 0$</p><p>$(x+1)^2 = 0$</p><p>$x+1 = 0$</p><p>$x = -1$</p><p>Значит, функция определена для всех действительных чисел, кроме $x=-1$.</p><p><strong>Ответ:</strong> $D(f) = (-\infty; -1) \cup (-1; +\infty)$.</p><p><strong>4)</strong> Преобразуем функцию $f(x) = (x+1)^{-\frac{3}{4}} + 3,5$ аналогично предыдущему пункту:</p><p>$f(x) = \frac{1}{(x+1)^{\frac{3}{4}}} + 3,5 = \frac{1}{\sqrt[4]{(x+1)^3}} + 3,5$</p><p>Здесь мы имеем дело с корнем четной степени (4-й степени) в знаменателе. Это накладывает два условия:</p><p>1. Выражение под корнем четной степени должно быть неотрицательным:</p><p>$(x+1)^3 \ge 0$</p><p>Это неравенство равносильно $x+1 \ge 0$, то есть $x \ge -1$.</p><p>2. Знаменатель не должен быть равен нулю:</p><p>$\sqrt[4]{(x+1)^3} \ne 0$</p><p>$(x+1)^3 \ne 0$</p><p>$x+1 \ne 0$, то есть $x \ne -1$.</p><p>Объединяя оба условия, $x \ge -1$ и $x \ne -1$, получаем строгое неравенство $x > -1$.</p><p><strong>Ответ:</strong> $D(f) = (-1; +\infty)$.</p>" ] #original: array:6 [ "id" => 1553194 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1051} "task" => array:2 [ "refs" => "1124978" "type" => "task" ] "text" => "<p><strong>1)</strong> Для нахождения области определения функции $f(x) = \frac{2}{(x-1)^2} + 4$ необходимо учесть, что знаменатель дроби не может быть равен нулю. В данном случае знаменатель — это $(x-1)^2$. Приравняем его к нулю, чтобы найти недопустимые значения $x$:</p><p>$(x-1)^2 = 0$</p><p>$x-1 = 0$</p><p>$x = 1$</p><p>Таким образом, функция определена для всех действительных чисел $x$, кроме $x=1$.</p><p><strong>Ответ:</strong> $D(f) = (-\infty; 1) \cup (1; +\infty)$.</p><p><strong>2)</strong> Для функции $f(x) = -\frac{3}{(x+2)^3} + 1,5$ область определения также находится из условия, что знаменатель не должен обращаться в нуль. Знаменатель здесь — $(x+2)^3$. Найдем значение $x$, при котором он равен нулю:</p><p>$(x+2)^3 = 0$</p><p>$x+2 = 0$</p><p>$x = -2$</p><p>Следовательно, область определения функции — это множество всех действительных чисел, за исключением $x=-2$.</p><p><strong>Ответ:</strong> $D(f) = (-\infty; -2) \cup (-2; +\infty)$.</p><p><strong>3)</strong> Преобразуем функцию $f(x) = (x+1)^{-\frac{2}{3}} - 2,5$, используя свойство степени с отрицательным показателем $a^{-n} = \frac{1}{a^n}$ и свойство степени с рациональным показателем $a^{\frac{m}{n}} = \sqrt[n]{a^m}$:</p><p>$f(x) = \frac{1}{(x+1)^{\frac{2}{3}}} - 2,5 = \frac{1}{\sqrt[3]{(x+1)^2}} - 2,5$</p><p>Корень нечетной степени (здесь — кубический) определен для любого значения подкоренного выражения. Однако, так как это выражение находится в знаменателе, оно не должно быть равно нулю.</p><p>$\sqrt[3]{(x+1)^2} = 0$</p><p>$(x+1)^2 = 0$</p><p>$x+1 = 0$</p><p>$x = -1$</p><p>Значит, функция определена для всех действительных чисел, кроме $x=-1$.</p><p><strong>Ответ:</strong> $D(f) = (-\infty; -1) \cup (-1; +\infty)$.</p><p><strong>4)</strong> Преобразуем функцию $f(x) = (x+1)^{-\frac{3}{4}} + 3,5$ аналогично предыдущему пункту:</p><p>$f(x) = \frac{1}{(x+1)^{\frac{3}{4}}} + 3,5 = \frac{1}{\sqrt[4]{(x+1)^3}} + 3,5$</p><p>Здесь мы имеем дело с корнем четной степени (4-й степени) в знаменателе. Это накладывает два условия:</p><p>1. Выражение под корнем четной степени должно быть неотрицательным:</p><p>$(x+1)^3 \ge 0$</p><p>Это неравенство равносильно $x+1 \ge 0$, то есть $x \ge -1$.</p><p>2. Знаменатель не должен быть равен нулю:</p><p>$\sqrt[4]{(x+1)^3} \ne 0$</p><p>$(x+1)^3 \ne 0$</p><p>$x+1 \ne 0$, то есть $x \ne -1$.</p><p>Объединяя оба условия, $x \ge -1$ и $x \ne -1$, получаем строгое неравенство $x > -1$.</p><p><strong>Ответ:</strong> $D(f) = (-1; +\infty)$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1124979" "type" => "task" ] "previous" => array:2 [ "refs" => "1124977" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\BookPage {#1103 #connection: "mysql" 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Illuminate\Database\Eloquent\Collection {#1049 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:1 [ 0 => App\Models\Element {#1109 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1263459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Edition {#1098} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1100035" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "57-1.jpg" "alt" => null "width" => "2009" "height" => 2912 "path" => "/media/algebra_11/abylkasymova1-u19/0-1/57-1.webp?ts=1752224242" ] ] ] #original: array:6 [ "id" => 1263459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110} "book_page" => array:2 [ "refs" => "1100035" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "57-1.jpg" "alt" => null "width" => "2009" "height" => 2912 "path" => "/media/algebra_11/abylkasymova1-u19/0-1/57-1.webp?ts=1752224242" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1100036" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1100034" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1184 #items: array:7 [ 0 => App\Models\Task {#1198 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: 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"id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1124975" "type" => "task" ] "previous" => array:2 [ "refs" => "1124973" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1100035" "type" => "book_page" ] ] #original: array:24 [ "id" => 1124974 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" 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[] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1241 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1124978 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/08-7" "field_display_title" => "8.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1248 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false } ] #original: array:24 [ "id" => 1124978 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/08-7" "field_display_title" => "8.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1108} "content" => Illuminate\Database\Eloquent\Collection {#1038} "next" => array:2 [ "refs" => "1124979" "type" => "task" ] "previous" => array:2 [ "refs" => "1124977" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1102} "page" => array:2 [ "refs" => "1100035" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№8.7 (с. 57)
Условие. №8.7 (с. 57)
скриншот условия
8.7. 1) $f(x) = \frac{2}{(x-1)^2} + 4;$
2) $f(x) = -\frac{3}{(x+2)^3} + 1,5;$
3) $f(x) = (x+1)^{-\frac{3}{2}} - 2,5;$
4) $f(x) = (x+1)^{-\frac{3}{4}} + 3,5.$
Решение 2 (rus). №8.7 (с. 57)
1) Для нахождения области определения функции $f(x) = \frac{2}{(x-1)^2} + 4$ необходимо учесть, что знаменатель дроби не может быть равен нулю. В данном случае знаменатель — это $(x-1)^2$. Приравняем его к нулю, чтобы найти недопустимые значения $x$:
$(x-1)^2 = 0$
$x-1 = 0$
$x = 1$
Таким образом, функция определена для всех действительных чисел $x$, кроме $x=1$.
Ответ: $D(f) = (-\infty; 1) \cup (1; +\infty)$.
2) Для функции $f(x) = -\frac{3}{(x+2)^3} + 1,5$ область определения также находится из условия, что знаменатель не должен обращаться в нуль. Знаменатель здесь — $(x+2)^3$. Найдем значение $x$, при котором он равен нулю:
$(x+2)^3 = 0$
$x+2 = 0$
$x = -2$
Следовательно, область определения функции — это множество всех действительных чисел, за исключением $x=-2$.
Ответ: $D(f) = (-\infty; -2) \cup (-2; +\infty)$.
3) Преобразуем функцию $f(x) = (x+1)^{-\frac{2}{3}} - 2,5$, используя свойство степени с отрицательным показателем $a^{-n} = \frac{1}{a^n}$ и свойство степени с рациональным показателем $a^{\frac{m}{n}} = \sqrt[n]{a^m}$:
$f(x) = \frac{1}{(x+1)^{\frac{2}{3}}} - 2,5 = \frac{1}{\sqrt[3]{(x+1)^2}} - 2,5$
Корень нечетной степени (здесь — кубический) определен для любого значения подкоренного выражения. Однако, так как это выражение находится в знаменателе, оно не должно быть равно нулю.
$\sqrt[3]{(x+1)^2} = 0$
$(x+1)^2 = 0$
$x+1 = 0$
$x = -1$
Значит, функция определена для всех действительных чисел, кроме $x=-1$.
Ответ: $D(f) = (-\infty; -1) \cup (-1; +\infty)$.
4) Преобразуем функцию $f(x) = (x+1)^{-\frac{3}{4}} + 3,5$ аналогично предыдущему пункту:
$f(x) = \frac{1}{(x+1)^{\frac{3}{4}}} + 3,5 = \frac{1}{\sqrt[4]{(x+1)^3}} + 3,5$
Здесь мы имеем дело с корнем четной степени (4-й степени) в знаменателе. Это накладывает два условия:
1. Выражение под корнем четной степени должно быть неотрицательным:
$(x+1)^3 \ge 0$
Это неравенство равносильно $x+1 \ge 0$, то есть $x \ge -1$.
2. Знаменатель не должен быть равен нулю:
$\sqrt[4]{(x+1)^3} \ne 0$
$(x+1)^3 \ne 0$
$x+1 \ne 0$, то есть $x \ne -1$.
Объединяя оба условия, $x \ge -1$ и $x \ne -1$, получаем строгое неравенство $x > -1$.
Ответ: $D(f) = (-1; +\infty)$.
Другие задания:
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