Quiz — Fiche méthodes - Fonctions réciproques.pdf

Question 1 sur 5 10:00

[{"id":85915,"question":"Quelle est la fonction réciproque de f(x) = e^x ?","option_a":"ln(x)","option_b":"e^x","option_c":"x","option_d":"1\/e^x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle e^x est bijective sur ℝ, et sa réciproque est la fonction logarithme népérien ln(x), définie sur ]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\": \"ln(x)\", \"b\": \"e^x\", \"c\": \"x\", \"d\": \"1\/e^x\"}}","_debug_options_count":4},{"id":85916,"question":"Si f(x) = 2x + 3, quelle est l'expression de sa fonction réciproque f⁻¹(x) ?","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, on résout y = 2x + 3 en x : x = (y - 3)\/2. On échange x et y pour obtenir 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":85917,"question":"La fonction f(x) = x² est-elle bijective sur ℝ ? Pourquoi ?","option_a":"Oui, car elle est toujours croissante","option_b":"Non, car elle n'est pas injective (f(2) = f(-2))","option_c":"Oui, car elle est surjective","option_d":"Non, car elle n'est pas continue","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction est bijective si elle est à la fois injective (chaque image a un unique antécédent) et surjective. f(x) = x² n'est pas injective sur ℝ car f(2) = f(-2) = 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Oui, car elle est toujours croissante\", \"b\": \"Non, car elle n'est","_debug_options_count":4},{"id":85918,"question":"Quelle est la dérivée de la fonction réciproque de f(x) = sin(x) pour x ∈ ]-π\/2, π\/2[ ?","option_a":"cos(x)","option_b":"1\/cos(x)","option_c":"-1\/cos²(x)","option_d":"1\/sin(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si f(x) = sin(x), alors f⁻¹(x) = arcsin(x). La dérivée de arcsin(x) est 1\/√(1 - x²), qui peut s'écrire 1\/cos(arcsin(x)) = 1\/cos(x) pour x ∈ ]-π\/2, π\/2[.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"cos(x)\", \"b\": \"1\/cos(x)\", \"c\": \"-1\/cos²(x)\", \"d\": \"1\/sin(x)\"}}","_debug_options_count":4},{"id":85919,"question":"Résoudre l'équation ln(x) = 2.","option_a":"x = e²","option_b":"x = 2","option_c":"x = ln(2)","option_d":"x = 1\/e²","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction ln est la réciproque de l'exponentielle. Donc ln(x) = 2 équivaut à x = e².","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 = e²\", \"b\": \"x = 2\", \"c\": \"x = ln(2)\", \"d\": \"x = 1\/e²\"}}","_debug_options_count":4}]