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{"name": "Functions: domain and range", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Functions: domain and range", "tags": [], "metadata": {"description": "<p>Determine the largest possible domain of a rational function.</p>", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "statement": "<p>Consider the function \\[ f(x) = \\frac{\\var{a}}{(\\simplify{(x+{b})(x^2+{c})}} \\]</p>\n<p></p>", "advice": "<p>For this problem, the function will not be defined if the denominator is zero.</p>\n<p>a) So, $f(x)$ is not defined if $\\simplify{x+{b}}=0\\implies x=\\simplify{-{b}}$.</p>\n<p>b) The largest possible domain is the set of real numbers $\\mathbb{R}$ excluding any numbers where $f$ is not defined. Therefore, the largest possible domain is</p>\n<p>$\\mathbb{R}\\backslash \\{\\simplify{-{b}}\\}$.&nbsp;</p>", "rulesets": {}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..15)", "description": "<p>Non-zero numbers&nbsp;</p>", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-15..15 except 0)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c"], "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>State the value of $x$ for which $f(x)$ is NOT defined.</p>\n<p>$x=$&nbsp;[[0]]</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "ansA", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "-{b}", "maxValue": "-{b}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": true, "customName": "(b)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The largest possible domain for $f$ is:</p>\n<p></p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$\\mathbb{R}\\backslash\\{\\var{c}\\}$", "$\\mathbb{R}$", "$\\mathbb{R}\\backslash\\{\\simplify{-{b}}\\}$", "$\\{\\simplify{-{b}}\\}$"], "matrix": [0, 0, "1", 0], "distractors": ["Incorrect answer!", "Incorrect answer!", "", "Incorrect answer!"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}]}]}], "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}]}