// Numbas version: exam_results_page_options {"name": "NCM4 FE-F2022 Day 2 Unit 2 Piecewise and Composite Functions", "metadata": {"description": "

This test will assess a students ability to work with piecewise and composite functions.

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Evaluate the following function at $x=\\var{xValue}$:

\\[ f(x) = \\simplify{ {a} x^2+ {b}x + {c}} \\]

$f(\\var{xValue})=$

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Evaluate the following function at $x=\\var{xValue}$:

\\[ f(x) = \\simplify{ {a} x^2+ {b}x + {c}} \\]

$f(\\var{xValue})=$

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Evaluate the following:

\\[ f( \\simplify{ {aSub}x+{bSub} } ) \\] given that \\[ f(x) = \\simplify{{a}x^2+{b}x+{c} }\\]

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Evaluate the following composition \\[ f( e^{\\simplify{{r}x}}) \\] given:

\\[ f(x) = \\simplify{ {a}x^2+{b}x+{c} } \\]

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GIVEN:

\\[ f(x) = \\simplify{{a}x} \\]

and

\\[ g(x) = e^{\\var{r}x} \\]

and

\\[ h(x) = \\simplify{ {b}x^2+{c} } \\]

FIND: The following compositions of functions

Recall that \\[ (f \\circ g)(x) = f(g(x)) \\]

\\[ (f \\circ g)(x) \\]

", "answer": "{a}*e^({r}*x)", "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": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

\\[ (g \\circ f)(x) \\]

", "answer": "e^({r}*{a}*x)", "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": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

\\[ (h \\circ f)(x) \\]

", "answer": "{a}^2*{b}*x^2+{c}", "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": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

\\[ (g \\circ h)(x) \\]

", "answer": "e^({r}*{b}*x^2+{r}*{c})", "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": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

\\[ (h \\circ g)(x) \\]

", "answer": "{b}*e^(2*{r}*x)+{c}", "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": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

\\[ (f \\circ h)(x) \\]

", "answer": "{a}*{b}*x^2+{a}*{c}", "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": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "NCM4 U2 Compose with Rational and Square Root Functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Terry Young", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3130/"}], "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "statement": "

GIVEN:

\\[ f(x) = \\frac{\\var{n}}{\\var{a}x^2+\\var{b}} \\]

and

\\[ g(x) = \\sqrt{ \\simplify{x + {c}}} \\]

FIND: The composition of functions indicated below.

\\[ (f \\circ g)(x) \\]

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\\[ (g \\circ f)(x) \\]

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\\[ (f \\circ g)(\\var{xVal}) \\]

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\\[ (g \\circ f)(\\var{xVal}) \\]

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GIVEN:

{table1}

and

{table2}

FIND: Evaluate the following function compositions.

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\\[ (f \\circ g)(1) \\]

", "minValue": "ansA", "maxValue": "ansA", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

\\[ (g \\circ f)(-1) \\]

", "minValue": "ansB", "maxValue": "ansB", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Math 3 Evaluate Piecewise Function M/C", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Terry Young", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3130/"}], "ungrouped_variables": ["xbound", "m1", "b1", "m2", "b2", "ansA", "ansB", "ansC", "ansD"], "functions": {}, "preamble": {"js": "", "css": ""}, "variables": {"xbound": {"group": "Ungrouped variables", "name": "xbound", "description": "", "definition": "random(-5..5)", "templateType": "anything"}, "ansA": {"group": "Ungrouped variables", "name": "ansA", "description": "", "definition": "m1*xbound+b1", "templateType": "anything"}, "ansD": {"group": "Ungrouped variables", "name": "ansD", "description": "", "definition": "-m2*xbound+b2", "templateType": "anything"}, "b1": {"group": "Ungrouped variables", "name": "b1", "description": "", "definition": "random(-10..10)", "templateType": "anything"}, "b2": {"group": "Ungrouped variables", "name": "b2", "description": "", "definition": "random(-10..10)", "templateType": "anything"}, "ansC": {"group": "Ungrouped variables", "name": "ansC", "description": "", "definition": "-m1*xbound+b1", "templateType": "anything"}, "m1": {"group": "Ungrouped variables", "name": "m1", "description": "", "definition": "random(-5..5 except 0)", "templateType": "anything"}, "m2": {"group": "Ungrouped variables", "name": "m2", "description": "", "definition": "random(-5..5 except [0,m1])", "templateType": "anything"}, "ansB": {"group": "Ungrouped variables", "name": "ansB", "description": "", "definition": "m2*xbound+b2", "templateType": "anything"}}, "parts": [{"displayType": "radiogroup", "maxMarks": 0, "showCorrectAnswer": true, "shuffleChoices": false, "unitTests": [], "customName": "", "marks": 0, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "type": "1_n_2", "customMarkingAlgorithm": "", "choices": ["$\\var{ansA}$", "$\\var{ansB}$", "$\\var{ansC}$", "$\\var{ansD}$"], "showFeedbackIcon": true, "showCellAnswerState": true, "useCustomName": false, "variableReplacementStrategy": "originalfirst", "minMarks": 0, "matrix": [0, "1", 0, 0], "scripts": {}, "displayColumns": 0, "extendBaseMarkingAlgorithm": true, "distractors": ["", "", "", ""]}], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "tags": [], "variable_groups": [], "advice": "", "metadata": {"licence": "All rights reserved", "description": ""}, "statement": "

GIVEN:

$\\simplify{f}(\\simplify{x})=\\left\\{\\begin{align}&\\simplify{{m1}*x+{b1}},&& \\text{ for } \\simplify{x}< \\var{xbound}, \\\\&\\simplify{{m2}x+{b2}},&& \\text{ for } {x}\\geq \\var{xbound}\\end{align}\\right.$

FIND: $f(\\var{xbound})$

"}, {"name": "Math 3 Evaluate Piecewise Function M/C", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Terry Young", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3130/"}], "ungrouped_variables": ["xbound", "m1", "b1", "m2", "b2", "ansA", "ansB", "ansC", "ansD"], "functions": {}, "preamble": {"js": "", "css": ""}, "variables": {"xbound": {"group": "Ungrouped variables", "name": "xbound", "description": "", "definition": "random(-5..5)", "templateType": "anything"}, "ansA": {"group": "Ungrouped variables", "name": "ansA", "description": "", "definition": "m1*xbound+b1", "templateType": "anything"}, "ansD": {"group": "Ungrouped variables", "name": "ansD", "description": "", "definition": "-m2*xbound+b2", "templateType": "anything"}, "b1": {"group": "Ungrouped variables", "name": "b1", "description": "", "definition": "random(-10..10)", "templateType": "anything"}, "b2": {"group": "Ungrouped variables", "name": "b2", "description": "", "definition": "random(-10..10)", "templateType": "anything"}, "ansC": {"group": "Ungrouped variables", "name": "ansC", "description": "", "definition": "-m1*xbound+b1", "templateType": "anything"}, "m1": {"group": "Ungrouped variables", "name": "m1", "description": "", "definition": "random(-5..5 except 0)", "templateType": "anything"}, "m2": {"group": "Ungrouped variables", "name": "m2", "description": "", "definition": "random(-5..5 except [0,m1])", "templateType": "anything"}, "ansB": {"group": "Ungrouped variables", "name": "ansB", "description": "", "definition": "m2*xbound+b2", "templateType": "anything"}}, "parts": [{"displayType": "radiogroup", "maxMarks": 0, "showCorrectAnswer": true, "shuffleChoices": false, "unitTests": [], "customName": "", "marks": 0, "adaptiveMarkingPenalty": 0, "variableReplacements": [], "type": "1_n_2", "customMarkingAlgorithm": "", "choices": ["$\\var{ansA}$", "$\\var{ansB}$", "$\\var{ansC}$", "$\\var{ansD}$"], "showFeedbackIcon": true, "showCellAnswerState": true, "useCustomName": false, "variableReplacementStrategy": "originalfirst", "minMarks": 0, "matrix": [0, "1", 0, 0], "scripts": {}, "displayColumns": 0, "extendBaseMarkingAlgorithm": true, "distractors": ["", "", "", ""]}], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "tags": [], "variable_groups": [], "advice": "", "metadata": {"licence": "All rights reserved", "description": ""}, "statement": "

GIVEN:

FIND: $f(\\var{xbound})$

"}, {"name": "NCM4 Piecewise Function Domain and Range", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Terry Young", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3130/"}], "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "statement": "

GIVEN: The function graphed below.

{geogebra_applet(\"https://www.geogebra.org/m/nxtkhrwf\", [[\"shift\",s],[\"xa2\",xa], [\"xb1\",xa]])}

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Write the domain of the function in interval notation. For $\\pm\\infty$, please write out \"-infinity\" or \"+infinity\".

", "answer": "(-infinity,+infinity)", "displayAnswer": "(-infinity,+infinity)", "matchMode": "exact"}, {"type": "patternmatch", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Write the range of the funciton in interval notation. For $\\pm\\infty$, please write out \"-infinity\" or \"+infinity\".

", "answer": "[-{s},+infinity)", "displayAnswer": "[-{s},+infinity)", "matchMode": "exact"}]}]}], "allowPrinting": false, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": false, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "warnifunattempted", "message": "

You have not entered your answer for this question.

"}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": false, "timeout": {"action": "warn", "message": "

Your 80 minutes have expired for this test.

"}, "timedwarning": {"action": "warn", "message": "

You have 5 minutes remaining to complete this test.

"}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "

Please complete each problem. Be sure to pay attention to any answer entry format directions so that you will not have your answer marked incorrecting for a formatting error.

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