Quiz — 6387359e5c0d3_énoncé_magazine14-Fonctions Réciproques.pdf

Question 1 sur 10 20:00

[{"id":11023,"question":"Soit f(x) = 2x + 3. Quelle est l'expression de sa fonction réciproque f⁻¹(x) ?","option_a":"A. f⁻¹(x) = (x - 3)\/2","option_b":"B. f⁻¹(x) = 2x - 3","option_c":"C. f⁻¹(x) = (x + 3)\/2","option_d":"D. f⁻¹(x) = 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. En échangeant x et y, on obtient 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\": \"A. f⁻¹(x) = (x - 3)\/2\", \"b\": \"B. f⁻¹(x) = 2x - 3\", \"c\": \"C.","_debug_options_count":4},{"id":11024,"question":"La fonction f(x) = x² admet-elle une fonction réciproque sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = x² n'est pas injective sur ℝ (par exemple, f(2) = f(-2) = 4), donc elle n'admet pas de fonction réciproque sur ℝ.","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":11025,"question":"Soit f(x) = eˣ. Quelle est l'expression de sa fonction réciproque ?","option_a":"A. f⁻¹(x) = ln(x)","option_b":"B. f⁻¹(x) = eˣ","option_c":"C. f⁻¹(x) = x","option_d":"D. f⁻¹(x) = 1\/eˣ","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle eˣ est bijective sur ℝ, et sa fonction réciproque est la fonction logarithme népérien : f⁻¹(x) = 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\": \"A. f⁻¹(x) = ln(x)\", \"b\": \"B. f⁻¹(x) = eˣ\", \"c\": \"C. f⁻¹","_debug_options_count":4},{"id":11026,"question":"Soit f(x) = x³. La fonction f est-elle bijective sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = x³ est strictement croissante sur ℝ, donc elle est injective. De plus, elle est surjective sur ℝ, donc elle est bijective et admet une fonction 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},{"id":11027,"question":"Soit f(x) = 1\/x. Quelle est l'expression de sa fonction réciproque f⁻¹(x) ?","option_a":"A. f⁻¹(x) = 1\/x","option_b":"B. f⁻¹(x) = x","option_c":"C. f⁻¹(x) = -x","option_d":"D. f⁻¹(x) = x²","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = 1\/x est sa propre réciproque car f(f(x)) = f(1\/x) = x. Donc f⁻¹(x) = 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\": \"A. f⁻¹(x) = 1\/x\", \"b\": \"B. f⁻¹(x) = x\", \"c\": \"C. f⁻¹(x) ","_debug_options_count":4},{"id":11028,"question":"La fonction f(x) = sin(x) admet-elle une fonction réciproque sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction sin(x) n'est pas injective sur ℝ (elle est périodique), donc elle n'admet pas de fonction réciproque sur ℝ. On peut en revanche définir une réciproque sur un intervalle où elle est bijective, comme [-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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":11029,"question":"Soit f(x) = √x. Quelle est l'expression de sa fonction réciproque f⁻¹(x) ?","option_a":"A. f⁻¹(x) = x²","option_b":"B. f⁻¹(x) = √x","option_c":"C. f⁻¹(x) = 1\/√x","option_d":"D. f⁻¹(x) = x","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour trouver f⁻¹, on résout y = √x en x : x = y². En échangeant x et y, on obtient f⁻¹(x) = x².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. f⁻¹(x) = x²\", \"b\": \"B. f⁻¹(x) = √x\", \"c\": \"C. f⁻¹(","_debug_options_count":4},{"id":11030,"question":"Soit f(x) = 3x - 5. La fonction f est-elle bijective sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = 3x - 5 est une fonction affine de coefficient directeur non nul, donc elle est strictement monotone (croissante) et bijective sur ℝ.","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":11031,"question":"Soit f(x) = x² + 1. Peut-on définir une fonction réciproque sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = x² + 1 n'est pas injective sur ℝ (par exemple, f(1) = f(-1) = 2), donc elle n'admet pas de fonction réciproque sur ℝ. On peut en revanche la définir sur [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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":11032,"question":"Soit f(x) = ln(x). Quelle est l'expression de sa fonction réciproque f⁻¹(x) ?","option_a":"A. f⁻¹(x) = eˣ","option_b":"B. f⁻¹(x) = ln(x)","option_c":"C. f⁻¹(x) = x","option_d":"D. f⁻¹(x) = 1\/ln(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction logarithme népérien ln(x) est bijective sur ]0, +∞[, et sa fonction réciproque est la fonction exponentielle : f⁻¹(x) = eˣ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. f⁻¹(x) = eˣ\", \"b\": \"B. f⁻¹(x) = ln(x)\", \"c\": \"C. f⁻¹","_debug_options_count":4}]