Quiz — 63b58c132ba22_corrigé_Top50_Bijection.pdf

Question 1 sur 10 20:00

[{"id":117871,"question":"Soit f: ℝ → ℝ définie par f(x) = 2x + 1. Cette fonction est-elle bijective ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f est bijective car elle est injective (f(a)=f(b) ⇒ a=b) et surjective (pour tout y, x=(y-1)\/2 existe).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":117872,"question":"Quelle est la fonction réciproque de f(x) = 3x - 2 ?","option_a":"A) f⁻¹(x) = (x + 2)\/3","option_b":"B) f⁻¹(x) = (x - 2)\/3","option_c":"C) f⁻¹(x) = 3x + 2","option_d":"D) f⁻¹(x) = x\/3 - 2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour trouver f⁻¹, on résout y = 3x - 2 ⇒ x = (y + 2)\/3. Ainsi, f⁻¹(x) = (x + 2)\/3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A) f⁻¹(x) = (x + 2)\/3\", \"b\": \"B) f⁻¹(x) = (x - 2)\/3\", \"c\": ","_debug_options_count":4},{"id":117873,"question":"Une fonction constante peut-elle être bijective ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction constante n'est pas injective (plusieurs antécédents pour une même image), donc elle ne peut pas être bijective.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":117874,"question":"Soit f: ℝ → ℝ définie par f(x) = x². Cette fonction est-elle bijective ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f n'est pas injective (f(2)=f(-2)=4) et n'est pas surjective (les négatifs n'ont pas d'antécédent).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":117875,"question":"Quelle propriété doit vérifier une fonction pour être bijective ?","option_a":"A) Injective uniquement","option_b":"B) Surjective uniquement","option_c":"C) Injective et surjective","option_d":"D) Ni injective ni surjective","option_e":"","option_f":"","bonne_reponse":"c","explication":"Une fonction est bijective si et seulement si elle est à la fois injective et surjective.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A) Injective uniquement\", \"b\": \"B) Surjective uniquement\", \"c\": \"","_debug_options_count":4},{"id":117876,"question":"Soit f: ℝ → ℝ définie par f(x) = x³. Cette fonction est-elle bijective ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f est bijective car elle est injective (strictement croissante) et surjective (tout réel a un antécédent).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":117877,"question":"Si f est une bijection de E vers F, alors f⁻¹ est une bijection de F vers E. Cette affirmation est-elle vraie ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f est bijective, alors f⁻¹ existe et est également bijective, car elle conserve les propriétés d'injectivité et de surjectivité.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":117878,"question":"Quelle est la condition nécessaire et suffisante pour qu'une fonction soit bijective ?","option_a":"A) Être continue","option_b":"B) Être dérivable","option_c":"C) Être injective et surjective","option_d":"D) Être strictement monotone","option_e":"","option_f":"","bonne_reponse":"c","explication":"Une fonction est bijective si et seulement si elle est injective et surjective, indépendamment de sa continuité ou dérivabilité.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A) Être continue\", \"b\": \"B) Être dérivable\", \"c\": \"C) Être in","_debug_options_count":4},{"id":117879,"question":"Soit f: ℝ → ℝ définie par f(x) = eˣ. Cette fonction est-elle bijective ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f est injective (strictement croissante) et surjective (tout réel positif a un antécédent), donc elle est bijective.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":117880,"question":"Si f est une bijection, alors f⁻¹(f(x)) = x pour tout x. Cette égalité est-elle toujours vraie ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, f⁻¹(f(x)) = x pour tout x si f est bijective, car f⁻¹ est l'application réciproque.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]