// Numbas version: exam_results_page_options
{"name": "Domain of a piecewise function", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Domain of a piecewise function", "tags": [], "metadata": {"description": "<p>Multiple choice <g class=\"gr_ gr_4 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar multiReplace\" id=\"4\" data-gr-id=\"4\">question</g>. Given a randomised polynomial select the possible ways of writing the domain of the function.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Given the real function below, you should be able to determine its&nbsp;domain.&nbsp;</p>", "advice": "<p>The function $\\simplify{{out}}$ is a piecewise function. It is defined on different parts (or pieces) of its domain by different (sub)functions.</p>\n<p></p>\n<p>In particular, the pieces of the domain are the intervals</p>\n<ul>\n<li>$\\simplify{{inp}}&lt; \\var{b0}$</li>\n<li>$\\var{b0}\\leq \\simplify{{inp}}\\leq&nbsp;\\var{b1}$,</li>\n<li>$\\var{b2}\\leq\\simplify{{inp}}&lt; \\var{b3}$,</li>\n<li>$\\var{b3}&lt;\\simplify{{inp}}\\leq \\var{b4}$ and</li>\n<li>$\\simplify{{inp}}\\ge\\var{b5}$.</li>\n</ul>\n<p>and their corresponding (sub)functions are given by the expressions</p>\n<ul>\n<li>$\\simplify{{p0}}$</li>\n<li>$\\simplify{{p1}}$</li>\n<li>$\\simplify{{p2}}$</li>\n<li>$\\simplify{{p3}}$</li>\n<li>$\\simplify{{p4}}$</li>\n</ul>\n<p></p>\n<p>In the case of the function $\\simplify{{out}}$ above, each of the (sub)functions are defined for the indicated intervals and so the domain of $\\simplify{{out}}$ are all of those intervals combined. The first and second interval join nicely without a gap&nbsp;unlike the rest and so we have \\[\\text{dom}(\\simplify{{out}})=\\{\\simplify{{inp}}\\in\\mathbb{R}:\\, \\simplify{{inp}}\\leq \\var{b1},\\; \\var{b2}\\leq \\simplify{{inp}}&lt;\\var{b3},\\; \\var{b3}&lt;\\simplify{{inp}}\\leq\\var{b4}, \\text{ or }\\; \\simplify{{inp}}\\ge \\var{b5}\\}.\\]</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"b4": {"name": "b4", "group": "Ungrouped variables", "definition": "b[4]", "description": "", "templateType": "anything", "can_override": false}, "p0": {"name": "p0", "group": "Ungrouped variables", "definition": "expression(pieces[0])", "description": "", "templateType": "anything", "can_override": false}, "p4": {"name": "p4", "group": "Ungrouped variables", "definition": "expression(pieces[4])", "description": "", "templateType": "anything", "can_override": false}, "c8": {"name": "c8", "group": 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except 0)", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "b[2]", "description": "", "templateType": "anything", "can_override": false}, "c6": {"name": "c6", "group": "Ungrouped variables", "definition": "random(-12..12 except 0)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["out", "inp", "b", "b0", "b1", "b2", "b3", "b4", "b5", "c1", "c2", "c3", "c4", "c5", "c6", "c7", "c8", "pieces", "p0", "p1", "p2", "p3", "p4"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "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>$\\simplify{{out}}(\\simplify{{inp}})=\\left\\{\\begin{align}&amp;\\simplify{{p0}},&amp;&amp; \\text{ for } \\simplify{{inp}}&lt; \\var{b0},&nbsp;\\\\&amp;\\simplify{{p1}},&amp;&amp; \\text{ for } \\var{b0}\\leq \\simplify{{inp}}\\leq&nbsp;\\var{b1},\\\\ &amp;\\simplify{{p2}},&amp;&amp; \\text{ for } \\var{b2}\\leq\\simplify{{inp}}&lt; \\var{b3},\\\\ &amp;\\simplify{{p3}},&amp;&amp; \\text{ for } \\var{b3}&lt;\\simplify{{inp}}\\leq \\var{b4},\\\\ &amp;\\simplify{{p4}},&amp;&amp; \\text{ for } \\simplify{{inp}}\\ge\\var{b5}.\\end{align}\\right.$</p>\n<p></p>\n<p>Which of the following is true of the piecewise function above?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": "1", "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["<p>$\\text{dom}(\\simplify{{out}})=\\mathbb{R}$</p>", "<p>$\\text{dom}(\\simplify{{out}})=\\{\\simplify{{inp}}\\in\\mathbb{R}:\\, \\simplify{{inp}}\\leq \\var{b1},\\; \\var{b2}\\leq \\simplify{{inp}}\\leq\\var{b4}, \\text{ or }\\; \\simplify{{inp}}\\ge \\var{b5}\\}$</p>", "<p>$\\text{dom}(\\simplify{{out}})=\\{\\simplify{{inp}}\\in\\mathbb{R}:\\, \\simplify{{inp}}\\leq \\var{b1},\\; \\var{b2}\\leq \\simplify{{inp}}&lt;\\var{b3},\\; \\var{b3}&lt;\\simplify{{inp}}\\leq\\var{b4}, \\text{ or }\\; \\simplify{{inp}}\\ge \\var{b5}\\}$</p>", "<p>$\\text{dom}(\\simplify{{out}})=\\{\\simplify{{inp}}\\in\\mathbb{R}:\\, \\simplify{{inp}}&lt; \\var{b0},\\; \\var{b2}\\leq \\simplify{{inp}}&lt;\\var{b3},\\; \\var{b3}&lt;\\simplify{{inp}}\\leq\\var{b4}, \\text{ or }\\; \\simplify{{inp}}\\ge \\var{b5}\\}$</p>", "<p>$\\text{dom}(\\simplify{{out}})=\\{\\simplify{{inp}}\\in\\mathbb{R}:\\, \\simplify{{inp}}&lt; \\var{b1},\\; \\var{b2}&lt; \\simplify{{inp}}\\leq\\var{b3},\\; \\var{b3}\\leq\\simplify{{inp}}&lt;\\var{b4}, \\text{ or }\\; \\simplify{{inp}}&gt; \\var{b5}\\}$</p>", "<p>$\\text{dom}(\\simplify{{out}})=\\mathbb{R}^5$</p>", "<p>$\\text{dom}(\\simplify{{out}})=\\{\\simplify{{inp}}\\in\\mathbb{R}:\\, \\simplify{{inp}}&lt; \\var{b0},\\; \\var{b0}&lt; \\simplify{{inp}}&lt;\\var{b1},\\;\\var{b2}\\leq \\simplify{{inp}}&lt;\\var{b3},\\; \\var{b3}&lt;\\simplify{{inp}}\\leq\\var{b4}, \\text{ or }\\; \\simplify{{inp}}\\ge \\var{b5}\\}$</p>"], "matrix": ["0", "0", "1", 0, 0, 0, 0], "distractors": ["", "", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}