تمارين الدوال العكسية - كتاب الرياضيات - الصف 12 - الفصل 1 - المملكة العربية السعودية

```json { "language": "ar", "direction": "rtl", "page_context": { "page_type": "exercises", "has_visual_elements": true }, "sections": [ { "order": 1, "type": "instruction", "content": "كون جدولاً للدالة f⁻¹ في كل مما يأتي إذا كانت موجودة، وإذا لم تكن موجودة، فاذكر السبب.", "associated_visual_elements": [] }, { "order": 2, "type": "exercise", "title": "36", "content": "جدول قيم للدالة f(x).", "associated_visual_elements": [0] }, { "order": 3, "type": "exercise", "title": "37", "content": "جدول قيم للدالة f(x).", "associated_visual_elements": [1] }, { "order": 4, "type": "exercise", "title": "38", "content": "درجات حرارة: تستعمل الدالة 32 + (9/5)x = f(x) للتحويل من درجات الحرارة السليزية إلى درجات الحرارة الفهرنهايتية، وتُستعمل الدالة (459.67 + x) (5/9) = k(x) للتحويل من درجات الحرارة الفهرنهايتية إلى درجات الحرارة المطلقة (كلفن).\na) أوجد f⁻¹، وماذا تمثل هذه الدالة؟\nb) أثبت أن كلاً من f, f⁻¹ دالة عكسية للأخرى، ومثل منحناهما على الشاشة نفسها في الحاسبة البيانية.\nc) أوجد (k ∘ f)(x)، وماذا تمثل هذه الدالة؟\nd) إذا كانت درجة الحرارة 60°C، فأوجد درجة الحرارة المطلقة المقابلة لها.", "associated_visual_elements": [] }, { "order": 5, "type": "instruction", "content": "ضع قيودًا على مجال كل دالة من الدوال الآتية حتى تصبح دالة متباينة. ثم أوجد الدالة العكسية لها:", "associated_visual_elements": [] }, { "order": 6, "type": "exercise", "title": "39", "content": "f(x) = (x-5)²", "associated_visual_elements": [2] }, { "order": 7, "type": "exercise", "title": "40", "content": "f(x) = 2 - |x+9|", "associated_visual_elements": [3] }, { "order": 8, "type": "exercise", "title": "41", "content": "f(x) = x² - 6", "associated_visual_elements": [4] }, { "order": 9, "type": "exercise", "title": "42", "content": "f(x) = |x+5| - 4", "associated_visual_elements": [5] }, { "order": 10, "type": "exercise", "title": "43", "content": "أزهار: تحتاج فاطمة إلى 75 زهرة لتزيين قاعة في إحدى المناسبات، فإذا كان بإمكانها شراء قرنفل بسعر 3 ريالات للزهرة الواحدة وشراء جوري بسعر 5 ريالات للزهرة الواحدة، فأجب عما يأتي:\na) اكتب دالة تمثل التكلفة الكلية لشراء الأزهار.\nb) أوجد الدالة العكسية لدالة التكلفة، وماذا يمثل كل متغير فيها؟\nc) حدد مجال دالة التكلفة، ومجال الدالة العكسية لها.\nd) إذا كانت التكلفة الكلية لشراء الأزهار 305 ريالات، فكم زهرة من القرنفل اشترت؟", "associated_visual_elements": [] }, { "order": 11, "type": "instruction", "content": "إذا كانت الدالة f⁻¹ موجودة، فاكتب المجال والمدى لكل من f, f⁻¹:", "associated_visual_elements": [] }, { "order": 12, "type": "exercise", "title": "44", "content": "f(x) = √x - 6", "associated_visual_elements": [] }, { "order": 13, "type": "exercise", "title": "45", "content": "f(x) = x² + 9", "associated_visual_elements": [] }, { "order": 14, "type": "exercise", "title": "46", "content": "f(x) = (3x + 1) / (x - 4)", "associated_visual_elements": [] }, { "order": 15, "type": "exercise", "title": "47", "content": "f(x) = (8x + 3) / (2x - 6)", "associated_visual_elements": [] }, { "order": 16, "type": "instruction", "content": "أوجد الدالة العكسية في كل مما يأتي، إن أمكن. ثم مثل f, f⁻¹ في مستوى إحداثي واحد. واذكر أية قيود على المجال:", "associated_visual_elements": [] }, { "order": 17, "type": "exercise", "title": "48", "content": "f(x) = { x² , -4 ≥ x ; -2x + 5 , -4 < x }", "associated_visual_elements": [] }, { "order": 18, "type": "exercise", "title": "49", "content": "f(x) = { -4x + 6 , -5 ≥ x ; 2x - 8 , -5 < x }", "associated_visual_elements": [] }, { "order": 19, "type": "exercise", "title": "50", "content": "اتصالات: أعلنت شركة لبيع أجهزة الهاتف المحمول عن عرض مبين في الشكل أدناه. فكانت الشركة تخصم 50 ريالاً وتمنح تخفيضًا مقداره 10% من سعر الجهاز الأصلي.\na) اكتب دالة r لسعر الجهاز بدلالة سعره الأصلي إذا تم خصم 50 ريالاً فقط.\nb) اكتب دالة d لسعر الجهاز بدلالة سعره الأصلي إذا تم منح التخفيض (10%) فقط.\nc) اكتب قاعدة تمثل T = r ∘ d إذا تم التخفيض ثم الخصم.\nd) أوجد T⁻¹، وماذا تمثل؟\ne) إذا كانت التكلفة الكلية لشراء الجهاز بعد التخفيض ثم الخصم 760 ريالاً، فكم يكون سعره الأصلي؟", "associated_visual_elements": [6] }, { "order": 20, "type": "instruction", "content": "إذا كانت f(x) = 8x - 4, g(x) = 2x + 6 فأوجد:", "associated_visual_elements": [] }, { "order": 21, "type": "exercise", "title": "51", "content": "(f⁻¹ ∘ g⁻¹)(x)", "associated_visual_elements": [] }, { "order": 22, "type": "exercise", "title": "52", "content": "(g⁻¹ ∘ f⁻¹)(x)", "associated_visual_elements": [] }, { "order": 23, "type": "exercise", "title": "53", "content": "(f ∘ g)⁻¹(x)", "associated_visual_elements": [] }, { "order": 24, "type": "exercise", "title": "54", "content": "(g ∘ f)⁻¹(x)", "associated_visual_elements": [] }, { "order": 25, "type": "footer", "content": "الفصل 1 تحليل الدوال", "associated_visual_elements": [] }, { "order": 26, "type": "page_number", "content": "72", "associated_visual_elements": [] }, { "order": 27, "type": "footer", "content": "وزارة التعليم\nMinistry of Education\n2025 - 1447", "associated_visual_elements": [] } ], "visual_elements": [ { "index": 0, "label": "Table 36", "question_number": "36", "type": "table", "location": "right side of page, top", "description": "A table showing x and f(x) values for a function.", "data": { "headers": ["x", "f(x)"], "rows": [ {"x": -6, "f(x)": -4}, {"x": -4, "f(x)": 0}, {"x": -1, "f(x)": 3}, {"x": 3, "f(x)": 5}, {"x": 6, "f(x)": 9}, {"x": 10, "f(x)": 13} ] } }, { "index": 1, "label": "Table 37", "question_number": "37", "type": "table", "location": "right side of page, second from top", "description": "A table showing x and f(x) values for a function.", "data": { "headers": ["x", "f(x)"], "rows": [ {"x": -3, "f(x)": 14}, {"x": -2, "f(x)": 11}, {"x": -1, "f(x)": 8}, {"x": 0, "f(x)": 10}, {"x": 1, "f(x)": 11}, {"x": 2, "f(x)": 16} ] } }, { "index": 2, "label": "Graph 39", "question_number": "39", "type": "graph", "location": "right side of page, top graph", "coordinate_system": "Standard Cartesian grid, 1 unit per square. Major grid lines are labeled every 4 units.", "shape": "Parabola opening upwards", "function": "f(x) = (x-5)²", "axes_ranges": {"x_min": -4, "x_max": 12, "y_min": -4, "y_max": 12}, "endpoints": [], "critical_points": [ { "type": "global_min", "coordinates": {"x": 5, "y": 0}, "description": "The vertex of the parabola, which is its global minimum. From origin (0,0): 5 squares right, 0 squares up/down." } ], "y_intercept": { "x": 0, "y": 25, "description": "The curve crosses the y-axis at (0, 25), which is outside the visible range of the graph." }, "end_behavior": { "left": "Arrow pointing up-left, indicating x→-∞, y→+∞", "right": "Arrow pointing up-right, indicating x→+∞, y→+∞" }, "key_points": [ {"x": 2, "y": 9, "description": "Point on the curve. From origin (0,0): 2 squares right, 9 squares up."}, {"x": 3, "y": 4, "description": "Point on the curve. From origin (0,0): 3 squares right, 4 squares up."}, {"x": 4, "y": 1, "description": "Point on the curve. From origin (0,0): 4 squares right, 1 square up."}, {"x": 6, "y": 1, "description": "Point on the curve. From origin (0,0): 6 squares right, 1 square up."}, {"x": 7, "y": 4, "description": "Point on the curve. From origin (0,0): 7 squares right, 4 squares up."}, {"x": 8, "y": 9, "description": "Point on the curve. From origin (0,0): 8 squares right, 9 squares up."} ], "description": "A parabola opening upwards, with its vertex at (5, 0). The curve extends infinitely upwards on both sides." }, { "index": 3, "label": "Graph 40", "question_number": "40", "type": "graph", "location": "right side of page, second graph from top", "coordinate_system": "Standard Cartesian grid, 1 unit per square. Major grid lines are labeled every 4 units.", "shape": "Inverted V-shape", "function": "f(x) = 2 - |x+9|", "axes_ranges": {"x_min": -16, "x_max": 0, "y_min": -12, "y_max": 4}, "endpoints": [], "critical_points": [ { "type": "global_max", "coordinates": {"x": -9, "y": 2}, "description": "The vertex of the inverted V-shape, which is its global maximum. From origin (0,0): 9 squares left, 2 squares up." } ], "y_intercept": { "x": 0, "y": -7, "description": "The curve crosses the y-axis at (0, -7). From origin (0,0): 0 squares left/right, 7 squares down." }, "end_behavior": { "left": "Arrow pointing down-left, indicating x→-∞, y→-∞", "right": "Arrow pointing down-right, indicating x→+∞, y→-∞" }, "key_points": [ {"x": -12, "y": -1, "description": "Point on the curve. From origin (0,0): 12 squares left, 1 square down."}, {"x": -11, "y": 0, "description": "Point on the curve. From origin (0,0): 11 squares left, 0 squares up/down."}, {"x": -10, "y": 1, "description": "Point on the curve. From origin (0,0): 10 squares left, 1 square up."}, {"x": -8, "y": 1, "description": "Point on the curve. From origin (0,0): 8 squares left, 1 square up."}, {"x": -7, "y": 0, "description": "Point on the curve. From origin (0,0): 7 squares left, 0 squares up/down."}, {"x": -6, "y": -1, "description": "Point on the curve. From origin (0,0): 6 squares left, 1 square down."} ], "description": "An inverted V-shaped graph with its vertex (global maximum) at (-9, 2). The curve extends infinitely downwards on both sides." }, { "index": 4, "label": "Graph 41", "question_number": "41", "type": "graph", "location": "right side of page, third graph from top", "coordinate_system": "Standard Cartesian grid, 1 unit per square. Major grid lines are labeled every 4 units.", "shape": "Parabola opening upwards", "function": "f(x) = x² - 6", "axes_ranges": {"x_min": -8, "x_max": 8, "y_min": -8, "y_max": 8}, "endpoints": [], "critical_points": [ { "type": "global_min", "coordinates": {"x": 0, "y": -6}, "description": "The vertex of the parabola, which is its global minimum. From origin (0,0): 0 squares left/right, 6 squares down." } ], "y_intercept": { "x": 0, "y": -6, "description": "The curve crosses the y-axis at (0, -6), which is also its global minimum." }, "end_behavior": { "left": "Arrow pointing up-left, indicating x→-∞, y→+∞", "right": "Arrow pointing up-right, indicating x→+∞, y→+∞" }, "key_points": [ {"x": -3, "y": 3, "description": "Point on the curve. From origin (0,0): 3 squares left, 3 squares up."}, {"x": -2, "y": -2, "description": "Point on the curve. From origin (0,0): 2 squares left, 2 squares down."}, {"x": -1, "y": -5, "description": "Point on the curve. From origin (0,0): 1 square left, 5 squares down."}, {"x": 1, "y": -5, "description": "Point on the curve. From origin (0,0): 1 square right, 5 squares down."}, {"x": 2, "y": -2, "description": "Point on the curve. From origin (0,0): 2 squares right, 2 squares down."}, {"x": 3, "y": 3, "description": "Point on the curve. From origin (0,0): 3 squares right, 3 squares up."} ], "description": "A parabola opening upwards, with its vertex (global minimum) at (0, -6). The curve extends infinitely upwards on both sides." }, { "index": 5, "label": "Graph 42", "question_number": "42", "type": "graph", "location": "right side of page, bottom graph", "coordinate_system": "Standard Cartesian grid, 1 unit per square. Major grid lines are labeled every 4 units.", "shape": "V-shape opening upwards", "function": "f(x) = |x+5| - 4", "axes_ranges": {"x_min": -12, "x_max": 0, "y_min": -4, "y_max": 8}, "endpoints": [], "critical_points": [ { "type": "global_min", "coordinates": {"x": -5, "y": -4}, "description": "The vertex of the V-shape, which is its global minimum. From origin (0,0): 5 squares left, 4 squares down." } ], "y_intercept": { "x": 0, "y": 1, "description": "The curve crosses the y-axis at (0, 1). From origin (0,0): 0 squares left/right, 1 square up." }, "end_behavior": { "left": "Arrow pointing up-left, indicating x→-∞, y→+∞", "right": "Arrow pointing up-right, indicating x→+∞, y→+∞" }, "key_points": [ {"x": -8, "y": -1, "description": "Point on the curve. From origin (0,0): 8 squares left, 1 square down."}, {"x": -7, "y": -2, "description": "Point on the curve. From origin (0,0): 7 squares left, 2 squares down."}, {"x": -6, "y": -3, "description": "Point on the curve. From origin (0,0): 6 squares left, 3 squares down."}, {"x": -4, "y": -3, "description": "Point on the curve. From origin (0,0): 4 squares left, 3 squares down."}, {"x": -3, "y": -2, "description": "Point on the curve. From origin (0,0): 3 squares left, 2 squares down."}, {"x": -2, "y": -1, "description": "Point on the curve. From origin (0,0): 2 squares left, 1 square down."} ], "description": "A V-shaped graph opening upwards, with its vertex (global minimum) at (-5, -4). The curve extends infinitely upwards on both sides." }, { "index": 6, "label": "Image 50", "question_number": "50", "type": "image", "location": "left side of page, middle", "description": "An advertisement for a mobile phone sales center. The title is 'مركز بيع الهواتف النقار' (Mobile Phone Sales Center). Below the title, it states '10% تخفيضات على سعر كل هاتف' (10% discounts on the price of every phone) and 'أو خصم 50 ريالاً عند البيع' (or a 50 Riyal discount upon sale). The image shows three smartphones: one large phone on the left, and two smaller phones on the right, all displaying various app icons on their screens." } ] } ```