{ "schemaVersion": "1.0", "moduleId": "calculus-functions-0580-pilot", "pathways": { "differentiation": { "minimumDistinctItems": 20, "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "items": [ { "id": "diff-q01", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = 2x^3. Work out dy/dx.", "data": { "family": "diff-deriv", "function": "2x^3", "derivative": "6x^2", "point": null, "ask": "deriv", "misconceptionDistractor": "2x^2", "misconceptionDistractors": [ { "value": "6x^3", "code": "cf-power-kept", "feedback": "Multiply by the power AND take 1 off it: the power must come down (axⁿ → anxⁿ⁻¹)." }, { "value": "12x^2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "6x^2", "difficulty": 1, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "Term by term (multiply by the power, take 1 off): dy/dx = 6x^2.", "misconceptionCode": "cf-power-not-multiplied", "options": [ "6x^2", "6x^3", "2x^2", "12x^2" ], "correctIndex": 0 }, { "id": "diff-q02", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = 5x^2. Work out dy/dx.", "data": { "family": "diff-deriv", "function": "5x^2", "derivative": "10x", "point": null, "ask": "deriv", "misconceptionDistractor": "10x^2", "misconceptionDistractors": [ { "value": "5x", "code": "cf-power-not-multiplied", "feedback": "The power rule has TWO moves: multiply by the power, THEN take 1 off the power. You only took 1 off." }, { "value": "20x", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "10x", "difficulty": 1, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "Term by term (multiply by the power, take 1 off): dy/dx = 10x.", "misconceptionCode": "cf-power-kept", "options": [ "5x", "10x", "10x^2", "20x" ], "correctIndex": 1 }, { "id": "diff-q03", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = 3x + 4. Work out dy/dx.", "data": { "family": "diff-deriv", "function": "3x + 4", "derivative": "3", "point": null, "ask": "deriv", "misconceptionDistractor": "4", "misconceptionDistractors": [ { "value": "3x", "code": "cf-power-kept", "feedback": "Multiply by the power AND take 1 off it: the power must come down (axⁿ → anxⁿ⁻¹)." }, { "value": "6", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "3", "difficulty": 1, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "Term by term (multiply by the power, take 1 off): dy/dx = 3.", "misconceptionCode": "cf-const-to-one", "options": [ "3x", "4", "3", "6" ], "correctIndex": 2 }, { "id": "diff-q04", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = x^3 − 2x. Work out dy/dx.", "data": { "family": "diff-deriv", "function": "x^3 − 2x", "derivative": "3x^2 − 2", "point": null, "ask": "deriv", "misconceptionDistractor": "x^2 − 2", "misconceptionDistractors": [ { "value": "3x^3 − 2x", "code": "cf-power-kept", "feedback": "Multiply by the power AND take 1 off it: the power must come down (axⁿ → anxⁿ⁻¹)." }, { "value": "6x^2 − 4", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "3x^2 − 2", "difficulty": 2, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "Term by term (multiply by the power, take 1 off): dy/dx = 3x^2 − 2.", "misconceptionCode": "cf-power-not-multiplied", "options": [ "3x^3 − 2x", "6x^2 − 4", "x^2 − 2", "3x^2 − 2" ], "correctIndex": 3 }, { "id": "diff-q05", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = 4x^3 + 3x^2 − 5. Work out dy/dx.", "data": { "family": "diff-deriv", "function": "4x^3 + 3x^2 − 5", "derivative": "12x^2 + 6x", "point": null, "ask": "deriv", "misconceptionDistractor": "12x^2 + 6x + 1", "misconceptionDistractors": [ { "value": "4x^2 + 3x", "code": "cf-power-not-multiplied", "feedback": "The power rule has TWO moves: multiply by the power, THEN take 1 off the power. You only took 1 off." }, { "value": "12x^3 + 6x^2", "code": "cf-power-kept", "feedback": "Multiply by the power AND take 1 off it: the power must come down (axⁿ → anxⁿ⁻¹)." } ] }, "correctAnswer": "12x^2 + 6x", "difficulty": 2, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "Term by term (multiply by the power, take 1 off): dy/dx = 12x^2 + 6x.", "misconceptionCode": "cf-const-to-one", "options": [ "12x^2 + 6x", "4x^2 + 3x", "12x^3 + 6x^2", "12x^2 + 6x + 1" ], "correctIndex": 0 }, { "id": "diff-q06", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = ½x^4. Work out dy/dx.", "data": { "family": "diff-deriv", "function": "½x^4", "derivative": "2x^3", "point": null, "ask": "deriv", "misconceptionDistractor": "2x^4", "misconceptionDistractors": [ { "value": "½x^3", "code": "cf-power-not-multiplied", "feedback": "The power rule has TWO moves: multiply by the power, THEN take 1 off the power. You only took 1 off." }, { "value": "4x^3", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "2x^3", "difficulty": 3, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "Term by term (multiply by the power, take 1 off): dy/dx = 2x^3.", "misconceptionCode": "cf-power-kept", "options": [ "2x^4", "2x^3", "½x^3", "4x^3" ], "correctIndex": 1 }, { "id": "diff-q07", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = −3x^2 + 6x. Work out dy/dx.", "data": { "family": "diff-deriv", "function": "−3x^2 + 6x", "derivative": "−6x + 6", "point": null, "ask": "deriv", "misconceptionDistractor": "−3x + 6", "misconceptionDistractors": [ { "value": "−6x^2 + 6x", "code": "cf-power-kept", "feedback": "Multiply by the power AND take 1 off it: the power must come down (axⁿ → anxⁿ⁻¹)." }, { "value": "−12x + 12", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "−6x + 6", "difficulty": 3, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "Term by term (multiply by the power, take 1 off): dy/dx = −6x + 6.", "misconceptionCode": "cf-power-not-multiplied", "options": [ "−3x + 6", "−6x^2 + 6x", "−6x + 6", "−12x + 12" ], "correctIndex": 2 }, { "id": "diff-q08", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = 2x^5 − x^2. Work out dy/dx.", "data": { "family": "diff-deriv", "function": "2x^5 − x^2", "derivative": "10x^4 − 2x", "point": null, "ask": "deriv", "misconceptionDistractor": "10x^5 − 2x^2", "misconceptionDistractors": [ { "value": "2x^4 − x", "code": "cf-power-not-multiplied", "feedback": "The power rule has TWO moves: multiply by the power, THEN take 1 off the power. You only took 1 off." }, { "value": "20x^4 − 4x", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "10x^4 − 2x", "difficulty": 4, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "Term by term (multiply by the power, take 1 off): dy/dx = 10x^4 − 2x.", "misconceptionCode": "cf-power-kept", "options": [ "2x^4 − x", "10x^5 − 2x^2", "20x^4 − 4x", "10x^4 − 2x" ], "correctIndex": 3 }, { "id": "diff-q09", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = 3x^2. Work out the gradient of the curve at x = 2.", "data": { "family": "diff-grad", "function": "3x^2", "derivative": "6x", "point": "2", "ask": "grad", "misconceptionDistractor": "13", "misconceptionDistractors": [ { "value": "6", "code": "cf-power-not-multiplied", "feedback": "The power rule has TWO moves: multiply by the power, THEN take 1 off the power. You only took 1 off." }, { "value": "11", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "12", "difficulty": 1, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "dy/dx = 6x; at x = 2 the gradient is 12.", "misconceptionCode": "cf-arithmetic", "options": [ "12", "6", "13", "11" ], "correctIndex": 0 }, { "id": "diff-q10", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = 2x^3 − 3x. Work out the gradient of the curve at x = 1.", "data": { "family": "diff-grad", "function": "2x^3 − 3x", "derivative": "6x^2 − 3", "point": "1", "ask": "grad", "misconceptionDistractor": "4", "misconceptionDistractors": [ { "value": "-1", "code": "cf-power-not-multiplied", "feedback": "The power rule has TWO moves: multiply by the power, THEN take 1 off the power. You only took 1 off." }, { "value": "2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "3", "difficulty": 2, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "dy/dx = 6x^2 − 3; at x = 1 the gradient is 3.", "misconceptionCode": "cf-arithmetic", "options": [ "-1", "3", "2", "4" ], "correctIndex": 1 }, { "id": "diff-q11", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = x^2 + 4x + 1. Work out the gradient of the curve at x = -1.", "data": { "family": "diff-grad", "function": "x^2 + 4x + 1", "derivative": "2x + 4", "point": "-1", "ask": "grad", "misconceptionDistractor": "3", "misconceptionDistractors": [ { "value": "1", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "-2", "code": "cf-power-kept", "feedback": "Multiply by the power AND take 1 off it: the power must come down (axⁿ → anxⁿ⁻¹)." } ] }, "correctAnswer": "2", "difficulty": 2, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "dy/dx = 2x + 4; at x = -1 the gradient is 2.", "misconceptionCode": "cf-arithmetic", "options": [ "1", "-2", "2", "3" ], "correctIndex": 2 }, { "id": "diff-q12", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = −x^2 + 6x. Work out the gradient of the curve at x = 4.", "data": { "family": "diff-grad", "function": "−x^2 + 6x", "derivative": "−2x + 6", "point": "4", "ask": "grad", "misconceptionDistractor": "2", "misconceptionDistractors": [ { "value": "-1", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "-3", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "-2", "difficulty": 3, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "dy/dx = −2x + 6; at x = 4 the gradient is -2.", "misconceptionCode": "cf-max-min-swapped", "options": [ "2", "-1", "-3", "-2" ], "correctIndex": 3 }, { "id": "diff-q13", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = x^2 − 4x + 3. Work out the x-coordinate of its stationary point.", "data": { "family": "diff-stat", "function": "x^2 − 4x + 3", "derivative": "2x − 4", "point": null, "ask": "stat", "misconceptionDistractor": "-1", "misconceptionDistractors": [ { "value": "3", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "-2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "2", "difficulty": 2, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "dy/dx = 2x − 4 = 0 gives x = 2.", "misconceptionCode": "cf-point-needs-y", "options": [ "2", "3", "-2", "-1" ], "correctIndex": 0 }, { "id": "diff-q14", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = x^3 − 3x. Work out the x-coordinate of its stationary point where x > 0.", "data": { "family": "diff-stat", "function": "x^3 − 3x", "derivative": "3x^2 − 3", "point": null, "ask": "stat", "misconceptionDistractor": "-2", "misconceptionDistractors": [ { "value": "2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "-1", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "1", "difficulty": 3, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "dy/dx = 3x^2 − 3 = 0 gives x = 1.", "misconceptionCode": "cf-point-needs-y", "options": [ "-2", "1", "2", "-1" ], "correctIndex": 1 }, { "id": "diff-q15", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = −x^2 + 4x + 5. Work out the x-coordinate of its stationary point.", "data": { "family": "diff-stat", "function": "−x^2 + 4x + 5", "derivative": "−2x + 4", "point": null, "ask": "stat", "misconceptionDistractor": "9", "misconceptionDistractors": [ { "value": "3", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "-2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "2", "difficulty": 3, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "dy/dx = −2x + 4 = 0 gives x = 2.", "misconceptionCode": "cf-point-needs-y", "options": [ "9", "3", "2", "-2" ], "correctIndex": 2 }, { "id": "diff-q16", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = −x^2 + 4x + 5. Work out the y-coordinate of its turning point.", "data": { "family": "diff-maxmin", "function": "−x^2 + 4x + 5", "derivative": "−2x + 4", "point": null, "ask": "maxmin", "misconceptionDistractor": "7", "misconceptionDistractors": [ { "value": "2", "code": "cf-point-needs-y", "feedback": "A stationary POINT needs both coordinates: put the x you found back into y to get the y." }, { "value": "10", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "9", "difficulty": 3, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "dy/dx = −2x + 4 = 0 at x = 2; y = 9 there.", "misconceptionCode": "cf-max-min-swapped", "options": [ "7", "2", "10", "9" ], "correctIndex": 3 }, { "id": "diff-q17", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = x^2 + 2x + 5. Work out the y-coordinate of its turning point.", "data": { "family": "diff-maxmin", "function": "x^2 + 2x + 5", "derivative": "2x + 2", "point": null, "ask": "maxmin", "misconceptionDistractor": "2", "misconceptionDistractors": [ { "value": "-1", "code": "cf-point-needs-y", "feedback": "A stationary POINT needs both coordinates: put the x you found back into y to get the y." }, { "value": "5", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "4", "difficulty": 3, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "dy/dx = 2x + 2 = 0 at x = -1; y = 4 there.", "misconceptionCode": "cf-max-min-swapped", "options": [ "4", "2", "-1", "5" ], "correctIndex": 0 }, { "id": "diff-q18", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "A curve has equation y = 2x^3 − 3x^2. Work out the y-coordinate of its turning point that is a minimum.", "data": { "family": "diff-maxmin", "function": "2x^3 − 3x^2", "derivative": "6x^2 − 6x", "point": null, "ask": "maxmin", "misconceptionDistractor": "0", "misconceptionDistractors": [ { "value": "1", "code": "cf-point-needs-y", "feedback": "A stationary POINT needs both coordinates: put the x you found back into y to get the y." }, { "value": "-2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "-1", "difficulty": 4, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "dy/dx = 6x^2 − 6x = 0 at x = 1; y = -1 there.", "misconceptionCode": "cf-max-min-swapped", "options": [ "1", "-1", "0", "-2" ], "correctIndex": 1 }, { "id": "diff-q19", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "The board shows a curve with a tangent drawn at a point. Along the tangent, a run of 1 gives a rise of 3. Estimate the gradient of the curve at that point.", "data": { "family": "diff-tangent", "function": "x^2", "derivative": "2x", "point": null, "ask": "tangent", "misconceptionDistractor": "1/3", "misconceptionDistractors": [ { "value": "4", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "-3", "code": "cf-max-min-swapped", "feedback": "Check the gradient EITHER SIDE of the point (or the second derivative): + to − is a maximum, − to + is a minimum." } ] }, "correctAnswer": "3", "difficulty": 1, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "gradient ≈ rise ÷ run = 3 ÷ 1 = 3.", "misconceptionCode": "cf-arithmetic", "options": [ "4", "1/3", "3", "-3" ], "correctIndex": 2 }, { "id": "diff-q20", "objectiveRecordIds": [ "0580-extended-E2.12-1", "0580-extended-E2.12-2", "0580-extended-E2.12-3", "0580-extended-E2.12-4" ], "prompt": "The board shows a curve with a tangent drawn at a point. Along the tangent, a run of 1 gives a rise of -1/2. Estimate the gradient of the curve at that point.", "data": { "family": "diff-tangent", "function": "x^2 − 2x", "derivative": "2x − 2", "point": null, "ask": "tangent", "misconceptionDistractor": "-3/2", "misconceptionDistractors": [ { "value": "-2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "1/2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "-1/2", "difficulty": 3, "hint": "The power rule has two moves per term: multiply by the power, then take 1 off it (axⁿ → anxⁿ⁻¹); a constant term becomes 0. The gradient at a point is dy/dx with the x put in. A stationary point has dy/dx = 0 — solve it, then put the x back into y for the full point. Maximum or minimum: check the gradient either side (+ to − is a maximum, − to + is a minimum), or the sign of the second derivative.", "explanation": "gradient ≈ rise ÷ run = -1/2 ÷ 1 = -1/2.", "misconceptionCode": "cf-max-min-swapped", "options": [ "-3/2", "-2", "1/2", "-1/2" ], "correctIndex": 3 } ] }, "functions": { "minimumDistinctItems": 20, "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "items": [ { "id": "func-q01", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 3x − 5. Work out f(4).", "data": { "family": "func-value", "f": "3x − 5", "g": null, "ask": "value", "misconceptionDistractor": "10", "misconceptionDistractors": [ { "value": "4", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "8", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "7", "difficulty": 1, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "f(4) = 3 × 4 + -5 = 7.", "misconceptionCode": "cf-arithmetic", "options": [ "7", "4", "10", "8" ], "correctIndex": 0 }, { "id": "func-q02", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 2x + 7. Work out f(-1).", "data": { "family": "func-value", "f": "2x + 7", "g": null, "ask": "value", "misconceptionDistractor": "7", "misconceptionDistractors": [ { "value": "3", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "6", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "5", "difficulty": 1, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "f(-1) = 2 × -1 + 7 = 5.", "misconceptionCode": "cf-arithmetic", "options": [ "3", "5", "7", "6" ], "correctIndex": 1 }, { "id": "func-q03", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = ½x + 3. Work out f(8).", "data": { "family": "func-value", "f": "½x + 3", "g": null, "ask": "value", "misconceptionDistractor": "15/2", "misconceptionDistractors": [ { "value": "13/2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "8", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "7", "difficulty": 2, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "f(8) = 1/2 × 8 + 3 = 7.", "misconceptionCode": "cf-arithmetic", "options": [ "13/2", "8", "7", "15/2" ], "correctIndex": 2 }, { "id": "func-q04", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = -2x + 6. Work out f(5).", "data": { "family": "func-value", "f": "-2x + 6", "g": null, "ask": "value", "misconceptionDistractor": "-6", "misconceptionDistractors": [ { "value": "-2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "-3", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "-4", "difficulty": 2, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "f(5) = -2 × 5 + 6 = -4.", "misconceptionCode": "cf-arithmetic", "options": [ "-2", "-3", "-6", "-4" ], "correctIndex": 3 }, { "id": "func-q05", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 3x − 5. Work out f⁻¹(4).", "data": { "family": "func-inverse-value", "f": "3x − 5", "g": null, "ask": "inverse-value", "misconceptionDistractor": "19/3", "misconceptionDistractors": [ { "value": "4", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "3", "difficulty": 1, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "Undo the machine: (4 − -5) ÷ 3 = 3.", "misconceptionCode": "cf-order-not-reversed", "options": [ "3", "4", "19/3", "2" ], "correctIndex": 0 }, { "id": "func-q06", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 2x + 3. Work out f⁻¹(7).", "data": { "family": "func-inverse-value", "f": "2x + 3", "g": null, "ask": "inverse-value", "misconceptionDistractor": "1/2", "misconceptionDistractors": [ { "value": "3", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "1", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "2", "difficulty": 2, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "Undo the machine: (7 − 3) ÷ 2 = 2.", "misconceptionCode": "cf-order-not-reversed", "options": [ "3", "2", "1", "1/2" ], "correctIndex": 1 }, { "id": "func-q07", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 5x − 4. Work out f⁻¹(6).", "data": { "family": "func-inverse-value", "f": "5x − 4", "g": null, "ask": "inverse-value", "misconceptionDistractor": "3", "misconceptionDistractors": [ { "value": "26/5", "code": "cf-order-not-reversed", "feedback": "An inverse reverses the steps AND their order: the LAST step of f is undone FIRST." }, { "value": "1", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "2", "difficulty": 2, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "Undo the machine: (6 − -4) ÷ 5 = 2.", "misconceptionCode": "cf-arithmetic", "options": [ "26/5", "1", "2", "3" ], "correctIndex": 2 }, { "id": "func-q08", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 4x + 9. Work out f⁻¹(1).", "data": { "family": "func-inverse-value", "f": "4x + 9", "g": null, "ask": "inverse-value", "misconceptionDistractor": "-35/4", "misconceptionDistractors": [ { "value": "-1", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "-3", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "-2", "difficulty": 3, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "Undo the machine: (1 − 9) ÷ 4 = -2.", "misconceptionCode": "cf-order-not-reversed", "options": [ "-35/4", "-1", "-3", "-2" ], "correctIndex": 3 }, { "id": "func-q09", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 3x − 5. Find f⁻¹(x).", "data": { "family": "func-inverse-expr", "f": "3x − 5", "g": null, "ask": "inverse-expr", "misconceptionDistractor": "x ÷ 3 + 5", "misconceptionDistractors": [ { "value": "⅓x − 5", "code": "cf-composite-order", "feedback": "gf(x) means g(f(x)): f acts FIRST, then g acts on f's output. The order matters." }, { "value": "(x − 5) ÷ 3", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "(x + 5) ÷ 3", "difficulty": 2, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "Reverse the steps in reverse order: f⁻¹(x) = (x + 5) ÷ 3.", "misconceptionCode": "cf-order-not-reversed", "options": [ "(x + 5) ÷ 3", "⅓x − 5", "(x − 5) ÷ 3", "x ÷ 3 + 5" ], "correctIndex": 0 }, { "id": "func-q10", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 2x + 6. Find f⁻¹(x).", "data": { "family": "func-inverse-expr", "f": "2x + 6", "g": null, "ask": "inverse-expr", "misconceptionDistractor": "x ÷ 2 − 6", "misconceptionDistractors": [ { "value": "½x + 6", "code": "cf-composite-order", "feedback": "gf(x) means g(f(x)): f acts FIRST, then g acts on f's output. The order matters." }, { "value": "(x + 6) ÷ 2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "(x − 6) ÷ 2", "difficulty": 3, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "Reverse the steps in reverse order: f⁻¹(x) = (x − 6) ÷ 2.", "misconceptionCode": "cf-order-not-reversed", "options": [ "x ÷ 2 − 6", "(x − 6) ÷ 2", "½x + 6", "(x + 6) ÷ 2" ], "correctIndex": 1 }, { "id": "func-q11", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 2x + 3 and g(x) = x + 2. Work out gf(4).", "data": { "family": "func-composite-value", "f": "2x + 3", "g": "x + 2", "ask": "composite-value", "misconceptionDistractor": "15", "misconceptionDistractors": [ { "value": "66", "code": "cf-composite-product", "feedback": "gf(x) is a CHAIN, not a product: feed f's output into g." }, { "value": "14", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "13", "difficulty": 2, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "f(4) = 11, then g(11) = 13.", "misconceptionCode": "cf-composite-order", "options": [ "66", "14", "13", "15" ], "correctIndex": 2 }, { "id": "func-q12", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 3x − 1 and g(x) = 2x + 5. Work out gf(2).", "data": { "family": "func-composite-value", "f": "3x − 1", "g": "2x + 5", "ask": "composite-value", "misconceptionDistractor": "26", "misconceptionDistractors": [ { "value": "45", "code": "cf-composite-product", "feedback": "gf(x) is a CHAIN, not a product: feed f's output into g." }, { "value": "16", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "15", "difficulty": 2, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "f(2) = 5, then g(5) = 15.", "misconceptionCode": "cf-composite-order", "options": [ "45", "16", "26", "15" ], "correctIndex": 3 }, { "id": "func-q13", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = x − 4 and g(x) = 3x + 1. Work out gf(2).", "data": { "family": "func-composite-value", "f": "x − 4", "g": "3x + 1", "ask": "composite-value", "misconceptionDistractor": "-14", "misconceptionDistractors": [ { "value": "3", "code": "cf-composite-order", "feedback": "gf(x) means g(f(x)): f acts FIRST, then g acts on f's output. The order matters." }, { "value": "-4", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "-5", "difficulty": 3, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "f(2) = -2, then g(-2) = -5.", "misconceptionCode": "cf-composite-product", "options": [ "-5", "-14", "3", "-4" ], "correctIndex": 0 }, { "id": "func-q14", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 2x + 1 and g(x) = ½x − 1. Work out gf(5).", "data": { "family": "func-composite-value", "f": "2x + 1", "g": "½x − 1", "ask": "composite-value", "misconceptionDistractor": "4", "misconceptionDistractors": [ { "value": "33/2", "code": "cf-composite-product", "feedback": "gf(x) is a CHAIN, not a product: feed f's output into g." }, { "value": "11/2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "9/2", "difficulty": 3, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "f(5) = 11, then g(11) = 9/2.", "misconceptionCode": "cf-composite-order", "options": [ "4", "9/2", "33/2", "11/2" ], "correctIndex": 1 }, { "id": "func-q15", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 2x + 3 and g(x) = x + 2. Express gf(x) in its simplest form.", "data": { "family": "func-composite-expr", "f": "2x + 3", "g": "x + 2", "ask": "composite-expr", "misconceptionDistractor": "2x + 7", "misconceptionDistractors": [ { "value": "(2x + 3)(x + 2)", "code": "cf-composite-product", "feedback": "gf(x) is a CHAIN, not a product: feed f's output into g." }, { "value": "2x + 1", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "2x + 5", "difficulty": 3, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "g(f(x)) = g(2x + 3) = 2x + 5.", "misconceptionCode": "cf-composite-order", "options": [ "2x + 7", "(2x + 3)(x + 2)", "2x + 5", "2x + 1" ], "correctIndex": 2 }, { "id": "func-q16", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 3x − 1 and g(x) = 2x. Express gf(x) in its simplest form.", "data": { "family": "func-composite-expr", "f": "3x − 1", "g": "2x", "ask": "composite-expr", "misconceptionDistractor": "(3x − 1)(2x)", "misconceptionDistractors": [ { "value": "6x − 1", "code": "cf-composite-order", "feedback": "gf(x) means g(f(x)): f acts FIRST, then g acts on f's output. The order matters." }, { "value": "7x − 2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "6x − 2", "difficulty": 3, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "g(f(x)) = g(3x − 1) = 6x − 2.", "misconceptionCode": "cf-composite-product", "options": [ "(3x − 1)(2x)", "6x − 1", "7x − 2", "6x − 2" ], "correctIndex": 3 }, { "id": "func-q17", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = ½x + 1 and g(x) = 4x − 2. Express gf(x) in its simplest form.", "data": { "family": "func-composite-expr", "f": "½x + 1", "g": "4x − 2", "ask": "composite-expr", "misconceptionDistractor": "2x", "misconceptionDistractors": [ { "value": "(½x + 1)(4x − 2)", "code": "cf-composite-product", "feedback": "gf(x) is a CHAIN, not a product: feed f's output into g." }, { "value": "2x + 6", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "2x + 2", "difficulty": 4, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "g(f(x)) = g(½x + 1) = 2x + 2.", "misconceptionCode": "cf-composite-order", "options": [ "2x + 2", "(½x + 1)(4x − 2)", "2x", "2x + 6" ], "correctIndex": 0 }, { "id": "func-q18", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 2x + 1 with domain {1, 2, 3}. Work out the LARGEST value in the range.", "data": { "family": "func-domain", "f": "2x + 1", "g": null, "ask": "domain", "misconceptionDistractor": "3", "misconceptionDistractors": [ { "value": "8", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "9", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "7", "difficulty": 2, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "The range is {3, 5, 7}; the largest value is 7.", "misconceptionCode": "cf-domain-range", "options": [ "8", "7", "9", "3" ], "correctIndex": 1 }, { "id": "func-q19", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = 3x − 2 with domain {0, 1, 2}. Work out the LARGEST value in the range.", "data": { "family": "func-domain", "f": "3x − 2", "g": null, "ask": "domain", "misconceptionDistractor": "2", "misconceptionDistractors": [ { "value": "-2", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "5", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "4", "difficulty": 3, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "The range is {-2, 1, 4}; the largest value is 4.", "misconceptionCode": "cf-domain-range", "options": [ "2", "-2", "4", "5" ], "correctIndex": 2 }, { "id": "func-q20", "objectiveRecordIds": [ "0580-extended-E2.13-1", "0580-extended-E2.13-2", "0580-extended-E2.13-3" ], "prompt": "f(x) = −x + 4 with domain {1, 2, 3}. Work out the LARGEST value in the range.", "data": { "family": "func-domain", "f": "−x + 4", "g": null, "ask": "domain", "misconceptionDistractor": "1", "misconceptionDistractors": [ { "value": "4", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." }, { "value": "5", "code": "cf-arithmetic", "feedback": "The method is right; check the arithmetic — keep fractions exact to the end." } ] }, "correctAnswer": "3", "difficulty": 3, "hint": "f(x) is a MACHINE: it multiplies, then adds. The inverse f⁻¹ undoes the machine — reverse the steps AND their order (the last step is undone first). gf(x) means g(f(x)): f acts first, then g acts on f's output — never f(x) × g(x). The domain is the inputs; the range is the outputs they give.", "explanation": "The range is {3, 2, 1}; the largest value is 3.", "misconceptionCode": "cf-domain-range", "options": [ "1", "4", "5", "3" ], "correctIndex": 3 } ] } } }