// Numbas version: exam_results_page_options
{"name": "inverse of a rational function", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "inverse of a rational function", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p></p>\n<p></p>", "advice": "<p>a)Domain of $f(x)$ is all real numbers except $\\frac{-\\var{d}}{\\var{c}}$.</p>\n<p>Let $y = \\frac{\\var{a}x \\ +\\ \\var{b}}{\\var{c}x\\ +\\ \\var{d}}$. Then&nbsp;</p>\n<p></p>\n<p>$y\\var{c}x \\ +\\ y\\var{d} = \\var{a}x\\ +\\ \\var{b}$&nbsp;</p>\n<p>$y\\var{c}x -\\var{a}x\\ = -y\\var{d} +\\var{b}$</p>\n<p>$x(y\\var{c} -\\var{a}) = -y\\var{d} +\\var{b}$</p>\n<p>$x = \\frac{-\\var{d}y +\\var{b}}{\\var{c}y -\\var{a}}$</p>\n<p></p>\n<p>Hence $f^{-1}(x) = \\frac{-\\var{d}x +\\var{b}}{\\var{c}x -\\var{a}}$</p>\n<p></p>\n<p>Domain of $f^{-1}(x)$ is all real numbers except $\\frac{\\var{a}}{\\var{c}}$</p>\n<p></p>\n<p>b) Domain of $f(x)$ is all real numbers except $\\frac{\\var{k}}{\\var{q}}$.</p>\n<p>Let $y = \\frac{\\var{s}x \\ -\\ \\var{t}}{\\var{q}x\\ -\\ \\var{k}}$. Then&nbsp;</p>\n<p></p>\n<p>$y\\var{q}x \\ -\\ y\\var{k} = \\var{s}x\\ -\\ \\var{t}$&nbsp;</p>\n<p>$y\\var{q}x -\\var{s}x\\ = y\\var{k} -\\var{t}$</p>\n<p>$x(y\\var{q} -\\var{s}) = y\\var{k} -\\var{t}$</p>\n<p>$x = \\frac{\\var{k}y -\\var{t}}{\\var{q}y -\\var{s}}$</p>\n<p></p>\n<p>Hence $f^{-1}(x) = \\frac{\\var{k}x -\\var{t}}{\\var{q}x -\\var{s}}$</p>\n<p></p>\n<p>Domain of $f^{-1}(x)$ is all real numbers except $\\frac{\\var{s}}{\\var{q}}$</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..15 except a)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..9 except a)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..15 except b except c)", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "random(1..15 except 0)", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(1..15 except s)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(1..15 except s)", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(1..15 except t except q)", "description": "", "templateType": "anything", "can_override": false}, "j": {"name": "j", "group": "Ungrouped variables", "definition": "random(1..15 except 0)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(1..15 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..15)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "s", "t", "q", "k", "j", "m"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "a)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Let $f(x) = \\frac{\\var{a}x \\ +\\ \\var{b}}{\\var{c}x\\ +\\ \\var{d}}$. Find the natural domain of $f$, $f^{-1}$ and the natural domain of $f^{-1}$.</p>\n<p>Domain of $f$ is all real numbers except [[1]]</p>\n<p>$f^{-1}(x) =$ [[0]]</p>\n<p>Domain of $f^{-1}$ is all real numbers except [[2]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(-{d}x + {b})/({c}x - {a})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(-{d})/{c}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}/{c}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Let $f(x) = \\frac{\\var{s}x \\ -\\ \\var{t}}{\\var{q}x\\ -\\ \\var{k}}$. Find the natural domain of $f$, $f^{-1}$ and the natural domain of $f^{-1}$.</p>\n<p>Domain of $f$ is all real numbers except [[0]]</p>\n<p>$f^{-1} =$ [[1]]</p>\n<p>Domain of $f^{-1}$ is all real numbers except [[2]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{k}/{q}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({k}x - {t})/({q}x -{s})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{s}/{q}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}]}]}], "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}]}