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{"name": "Functions: injective", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Functions: injective", "tags": [], "metadata": {"description": "<p>Gives students an introduction to the injective property of functions</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>Informally, when a function has exactly one input value for every output value we say it is <strong>injective</strong> or <strong>one-to-one</strong>.</p>\n<p>Formally, a function $f(x)$ is said to be injective when $f(x_1) = f(x_2)$ implies that $x_1=x_2$.</p>", "advice": "<p>Part a shows that to prove a function is not injective, find specific values of $x_1,x_2$ such that $f(x_1) = f(x_2)$ but $x_1 \\neq x_2$.</p>\n<p>Part b, this same method can be used to show that</p>\n<ul>\n<li>$\\sin$ is not injective since $\\sin(0) = \\sin(\\pi)$ but $0 \\neq \\pi$.</li>\n<li>$g(x)$ is not injective since $g(-1) = g(1)$ but $-1 \\neq 1$.</li>\n</ul>\n<p>There are two common methods to show a function is injective.</p>\n<ul>\n<li>If the function is continuous and has strictly positive (or strictly negative) derivative, then the function is injective. This is useful to prove that $\\tan(x)$ is injective because it is continuous on $(-\\pi/2,\\pi/2)$ and has derivative $\\sec^2(x) &gt; 0$.</li>\n<li>For some simple functions it is possible to prove the injective property directly. For example if you assume</li>\n</ul>\n<p style=\"text-align: center;\">$f(x_1) = f(x_2)$</p>\n<p>by definition of $f$</p>\n<p style=\"text-align: center;\">$2x_1 = 2x_2$</p>\n<p>and dividing by $2$:</p>\n<p style=\"text-align: center;\">$x_1=x_2$.</p>\n<p>Hence $f(x) = 2x$ is injective.</p>", "rulesets": {}, "extensions": [], "variables": {"x1": {"name": "x1", "group": "Ungrouped variables", "definition": "decimal(random(1..9) + random(1..9)/10)", "description": "", "templateType": "anything"}, "floorx1": {"name": "floorx1", "group": "Ungrouped variables", "definition": "floor(x1)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x1", "floorx1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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": "<p>Evaluate $\\lfloor \\var{x1} \\rfloor$ </p>", "minValue": "{floorx1}", "maxValue": "{floorx1}", "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": {"mark": {"script": "// extract question variables\nvar variables = this.question.scope.variables;\nvar unwrap = Numbas.jme.unwrapValue;\nvar x1 = unwrap(variables.x1);\nvar floorx1 = unwrap(variables.floorx1);\n\n// get the student's answers to the gaps\nvar ans = parseFloat(this.studentAnswer);\nvar floorans = Math.floor(ans);\n\nif (ans == x1) {\n    this.setCredit(0,\"You must choose a different number than $\" + x1 + \"$.\");\n} else if (floorans == floorx1) {\n    this.setCredit(1,\"Correct.\" ); \n}", "order": "instead"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find a value of $x$, other than $\\var{x1}$, such that $f(x) = \\var{floorx1}$</p>", "minValue": "{floorx1}", "maxValue": "{floorx1 + 0.9999999}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "1_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>Hence $\\lfloor \\,\\, \\rfloor : \\mathbb R \\mapsto \\mathbb Z$ is</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["Injective", "Not Injective"], "matrix": [0, "1"], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}]}]}], "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}]}