Quiz — Lycée Pilote Sfax - Résumé réciproque.pdf

Question 1 sur 10 20:00

[{"id":35610,"question":"Quelle est la condition nécessaire pour qu'une fonction admette une réciproque ?","option_a":"Être continue","option_b":"Être dérivable","option_c":"Être bijective","option_d":"Être croissante","option_e":"","option_f":"","bonne_reponse":"c","explication":"Une fonction doit être bijective (injective et surjective) pour admettre une réciproque définie sur tout son ensemble d'arrivée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Être continue\", \"b\": \"Être dérivable\", \"c\": \"Être bijective\",","_debug_options_count":4},{"id":35611,"question":"Si f(x) = 2x + 3, alors sa fonction réciproque f⁻¹(x) est égale à :","option_a":"(x - 3)\/2","option_b":"(x + 3)\/2","option_c":"2x - 3","option_d":"x\/2 + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour trouver f⁻¹(x), on résout y = 2x + 3 en x : x = (y - 3)\/2, donc f⁻¹(x) = (x - 3)\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"(x - 3)\/2\", \"b\": \"(x + 3)\/2\", \"c\": \"2x - 3\", \"d\": \"x\/2 + 3\"}}","_debug_options_count":4},{"id":35612,"question":"La réciproque d'une fonction strictement croissante est toujours strictement croissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si une fonction est strictement croissante, sa réciproque est également strictement croissante, car elle conserve l'ordre des images.","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":35613,"question":"Quelle est la réciproque de la fonction f(x) = eˣ ?","option_a":"ln(x)","option_b":"eˣ","option_c":"1\/eˣ","option_d":"x²","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle eˣ admet pour réciproque la fonction logarithme népérien ln(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"ln(x)\", \"b\": \"eˣ\", \"c\": \"1\/eˣ\", \"d\": \"x²\"}}","_debug_options_count":4},{"id":35614,"question":"Si f(g(x)) = x pour tout x, alors g est la réciproque de f.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f(g(x)) = x pour tout x dans le domaine de g, alors g est bien la réciproque de f, à condition que f soit 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":35615,"question":"Quelle est la réciproque de la fonction f(x) = √x (pour x ≥ 0) ?","option_a":"x²","option_b":"1\/√x","option_c":"x","option_d":"√(x+1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction racine carrée √x admet pour réciproque la fonction carrée x², définie pour x ≥ 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x²\", \"b\": \"1\/√x\", \"c\": \"x\", \"d\": \"√(x+1)\"}}","_debug_options_count":4},{"id":35616,"question":"La réciproque d'une fonction paire est toujours paire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction paire n'est généralement pas bijective (sauf si elle est définie sur un domaine restreint), donc elle n'admet pas toujours de réciproque.","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":35617,"question":"Si f(x) = x³, alors sa réciproque f⁻¹(x) est :","option_a":"x³","option_b":"∛x","option_c":"1\/x³","option_d":"x^(1\/3)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction cube x³ admet pour réciproque la fonction racine cubique ∛x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"x³\", \"b\": \"∛x\", \"c\": \"1\/x³\", \"d\": \"x^(1\/3)\"}}","_debug_options_count":4},{"id":35618,"question":"La réciproque d'une fonction strictement décroissante est toujours strictement décroissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si une fonction est strictement décroissante, sa réciproque est également strictement décroissante, car elle inverse l'ordre des images.","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":35619,"question":"Quelle est la réciproque de la fonction f(x) = 1\/x (pour x ≠ 0) ?","option_a":"x","option_b":"1\/x","option_c":"-x","option_d":"x²","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = 1\/x est sa propre réciproque, car f(f(x)) = x pour tout x ≠ 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"x\", \"b\": \"1\/x\", \"c\": \"-x\", \"d\": \"x²\"}}","_debug_options_count":4}]