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Multiple choice
Given the real function below, you should be able to determine its domain.
\n$\\simplify{f}(\\simplify{t})=\\left\\{\\begin{align}&\\simplify{0},&& \\text{ for } \\simplify{x}< \\var{b0}, \\\\&\\simplify{{a}x+{b}},&& \\text{ for } \\var{b0}\\leq \\simplify{x}< \\var{b1},\\\\ &\\simplify{{c}x^2+{d}},&& \\text{ for } \\var{b1}\\leq\\simplify{x}< \\var{b2},\\end{align}\\right.$
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\nWhat is the value of $f(\\var{Input2})$? [[1]]
", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false, "type": "gapfill", "scripts": {}, "marks": 0, "unitTests": [], "gaps": [{"showCorrectAnswer": true, "expectedVariableNames": [], "showPreview": true, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "type": "jme", "checkingAccuracy": 0.001, "answer": "{Answer1}", "checkVariableNames": false, "failureRate": 1, "scripts": {}, "marks": 1, "vsetRange": [0, 1], "unitTests": [], "vsetRangePoints": 5, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}, {"showCorrectAnswer": true, "expectedVariableNames": [], "showPreview": true, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "type": "jme", "checkingAccuracy": 0.001, "answer": "{Answer2}", "checkVariableNames": false, "failureRate": 1, "scripts": {}, "marks": 1, "vsetRange": [0, 1], "unitTests": [], "vsetRangePoints": 5, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}], "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "showFeedbackIcon": true}], "name": "Joshua's copy of Domain of a piecewise function", "rulesets": {}, "functions": {}, "preamble": {"js": "", "css": ""}, "variables": {"b2": {"definition": "b1+3", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b2"}, "Answer1": {"definition": "{a}*{Input1}+{b}", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "Answer1"}, "c": {"definition": "random(-2..2 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c"}, "a": {"definition": "random(2..5)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a"}, "b1": {"definition": "b0+2", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b1"}, "Input1": {"definition": "b0+1", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "Input1"}, "Input2": {"definition": "{b2}+4", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "Input2"}, "d": {"definition": "random(2..8)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "d"}, "b0": {"definition": "random(1..3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b0"}, "b": {"definition": "random(1..3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b"}, "Answer2": {"definition": "{c}*({b2}+4)^2+{d}", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "Answer2"}}, "advice": "The function $\\simplify{{out}}$ is a piecewise function. It is defined on different parts (or pieces) of its domain by different (sub)functions.
\n\nIn particular, the pieces of the domain are the intervals
\nand their corresponding (sub)functions are given by the expressions
\nIn 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 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}}<\\var{b3},\\; \\var{b3}<\\simplify{{inp}}\\leq\\var{b4}, \\text{ or }\\; \\simplify{{inp}}\\ge \\var{b5}\\}.\\]
\n", "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Joshua Calcutt", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3267/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Joshua Calcutt", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3267/"}]}