// Numbas version: finer_feedback_settings
{"name": "CA6 Inverse functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "CA6 Inverse functions", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>If $f(x) = \\dfrac{\\simplify{{m}x + {c}}}{\\simplify{x + {a}}}$, $x \\neq \\simplify{{-a}}$, find the inverse function $f^{-1}(x)$ v\u1edbi $c = \\var{c}, a = \\var{a}, m = \\var{m}$</p>", "advice": "<p>To find $f^{-1}x$, it can help to first set $f(x)$ to a different variable, which we will call $y$</p>\n<p style=\"text-align: center;\">$y = f(x) = \\dfrac{\\simplify{{m}x + {c}}}{\\simplify{x + {a}}}$</p>\n<p style=\"text-align: left;\">Since the function $f(x)$ take us from $x$ to $y$, the inverse function $f^{-1}$ will take us from $y$ to $x$. So to obtain $f^{-1}$, we want to rearrange $y = f(x) = \\dfrac{\\simplify{{m}x + {c}}}{\\simplify{x + {a}}}$ so that it's $x$ as a function of $y$.</p>\n<p style=\"text-align: center;\">$y = \\dfrac{\\simplify{{m}x + {c}}}{\\simplify{x + {a}}}$</p>\n<p style=\"text-align: center;\">$\\simplify{(x + {a})y} = \\simplify{{m}x + {c}}$</p>\n<p style=\"text-align: center;\">$\\simplify{xy + {a}y} = \\simplify{{m}x + {c}}$</p>\n<p style=\"text-align: center;\">$\\simplify{xy - {m}x} = \\simplify{{c} - {a}y}$</p>\n<p style=\"text-align: center;\">$\\simplify{x(y - {m})} = \\simplify{{c} - {a}y}$</p>\n<p style=\"text-align: center;\">$x = \\dfrac{\\simplify{{c} - {a}y}}{\\simplify{y - {m}}}$</p>\n<p style=\"text-align: left;\">Hence, $f^{-1}(y) = \\dfrac{\\var{c} - \\var{a}y}{y - (\\var{m})}$, and therefore</p>\n<p style=\"text-align: center;\">$f^{-1}(x) = \\dfrac{\\simplify{{c} - {a}y}}{\\simplify{y - {m}}}$</p>\n<p style=\"text-align: left;\">(Note: The inverse is valid for all values of $x$ except $x = \\var{m}$, since this would make the denominator equal to 0.)</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "abs(m)-abs(c)>0 or abs(m)-abs(c)<0", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "m"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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>$f^{-1}(x) = $ [[0]] &nbsp;</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": "({c}-{a}x)/(x-{m})", "answerSimplification": "fractionNumbers", "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": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Nguyen Hai Anh", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/30342/"}], "resources": []}]}], "contributors": [{"name": "Nguyen Hai Anh", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/30342/"}]}