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{"name": "Functions: Inverse Functions 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Functions: Inverse Functions 3", "tags": [], "metadata": {"description": "<p>Finding the inverse of a function of the form $f(x)=\\frac{x^2}{a}+b$ for $x \\geq 0$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>If $f(x)=\\simplify{x^2/{a}+{b}}$ for $x \\geq 0$, find the inverse function, $f^{-1}(x)$.</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>\\[ y = f(x) = \\simplify{x^2/{a}+{b}}\\]</p>\n<p>Since the function $f(x)$ takes 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 $y=\\simplify{x^2/{a}+{b}}$ so that it is $x$ as a function of $y$:</p>\n<p>\\[ \\begin{split} y &amp;\\,= \\simplify{x^2/{a}+{b}} \\\\ \\simplify{y-{b}} &amp;\\,= \\simplify{x^2/{a}} \\\\&nbsp;\\simplify{{a}(y-{b})} &amp;\\,= x^2 \\\\ \\implies x &amp;\\,=\\sqrt{\\simplify{{a}(y-{b})}}. \\end{split} \\]</p>\n<p>Hence, $f^{-1}(y) =\\sqrt{\\simplify{{a}(y-{b})}}$, and therefore \\[ f^{-1}(x) =\\sqrt{\\simplify{{a}(x-{b})}}.\\]</p>\n<p>(<strong>Note:</strong>&nbsp;The inverse is valid for all values of $x\\geq\\var{b}$.)</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "abs(a)>abs(b) or abs(b)>abs(a)", "maxRuns": 100}, "ungrouped_variables": ["a", "b"], "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]]</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": "sqrt({a}(x-{b}))", "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": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}