// Numbas version: exam_results_page_options {"name": "Domain of a rational 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 rational function", "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "

Given a randomised rational function select the possible ways of writing the domain of the function.

"}, "extensions": [], "preamble": {"js": "", "css": ""}, "statement": "

Given the real functions below, you should be able to determine their domains.

", "variable_groups": [], "functions": {}, "variables": {"b": {"name": "b", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "a+random(1..12)"}, "rat": {"name": "rat", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "if(n=3,'\\\\[\\\\simplify{{out}({inp})=({num[0]}{inp}+{num[1]})/(({inp}-{c[0]})({inp}-{c[1]})*({inp}-{c[2]}))}\\\\]',\nif(n=2,'\\\\[\\\\simplify{{out}({inp})=({num[0]}{inp}+{num[1]})/(({inp}-{c[0]})({inp}-{c[1]}))}\\\\]',\n'\\\\[\\\\simplify{{out}({inp})=({num[0]}{inp}+{num[1]})/(({inp}-{c[0]})({inp}-{c[1]})*({inp}-{c[2]})*({inp}-{c[3]}))}\\\\]'))\n \n \n"}, "set_of_holes": {"name": "set_of_holes", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "set(c)"}, "num": {"name": "num", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "[random(-12..12),random(-12..12 except 0)]"}, "n": {"name": "n", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(2..4)"}, "out": {"name": "out", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "expression(random('f','h','g','p','q','y'))"}, "inp": {"name": "inp", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "expression(random('x','r','s','t','w'))"}, "c": {"name": "c", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "sort(shuffle(-12..12)[0..n])"}, "a": {"name": "a", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(-12..-1)"}}, "tags": [], "parts": [{"variableReplacements": [], "gaps": [{"correctAnswerStyle": "plain", "variableReplacements": [], "correctAnswerFraction": false, "minValue": "0", "type": "numberentry", "maxValue": "0", "useCustomName": false, "mustBeReduced": false, "showFractionHint": true, "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "allowFractions": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 1}], "type": "gapfill", "useCustomName": false, "prompt": "

There is a single real number that is not in the domain of the function

\\[f(x)=\\frac{1}{x}.\\]

That number is [[0]].

", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}, {"variableReplacements": [], "gaps": [{"correctAnswerStyle": "plain", "variableReplacements": [], "correctAnswerFraction": false, "minValue": "{length(c)}", "type": "numberentry", "maxValue": "{length(c)}", "useCustomName": false, "mustBeReduced": false, "showFractionHint": true, "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "allowFractions": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 1}, {"checkingAccuracy": 0.001, "variableReplacements": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "valuegenerators": [], "type": "jme", "useCustomName": false, "failureRate": 1, "checkingType": "absdiff", "checkVariableNames": false, "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showPreview": true, "answer": "{set_of_holes}", "showCorrectAnswer": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 1}], "type": "gapfill", "useCustomName": false, "prompt": "

There are [[0]] real numbers that are not in the domain of {rat} these are [[1]].

Note: If the numbers were $-2,1$ and $4$ you would enter set(-2,1,4)

$\\mathbb{R}$

", "

$\\{x\\in \\mathbb{R}:\\,\\var{a+b}<x<\\var{a*b}\\}$

", "

$\\{x\\in \\mathbb{R}:\\,x\\ne \\var{c[0]},\\var{c[1]}\\}$

", "

$\\{x\\in \\mathbb{R}:\\,x\\ne \\var{a},\\var{b}\\}$

", "

$\\mathbb{R}\\setminus \\{\\var{a},\\var{b}\\}$

", "

$\\mathbb{R}\\setminus \\{\\var{c[0]},\\var{c[1]}\\}$

", "

$\\{x\\in \\mathbb{R}:\\,x\\ne \\var{-a},\\var{-b}\\}$

"], "unitTests": [], "showCellAnswerState": true, "displayType": "checkbox", "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "distractors": ["", "", "", "", "", "", ""], "displayColumns": "1", "warningType": "none", "type": "m_n_2", "useCustomName": false, "maxAnswers": 0, "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "prompt": "

Which of the following represents the domain of \\[\\simplify{g(x)=(x^2-{c[0]+c[1]}x+{c[0]*c[1]})/(x^2-{a+b}x+{a*b})}?\\]

", "minMarks": 0, "shuffleChoices": true, "customMarkingAlgorithm": ""}, {"variableReplacements": [], "matrix": ["1", 0], "showCorrectAnswer": true, "maxMarks": 0, "choices": ["

$\\{r\\in\\mathbb{R}:\\,r\\ne \\var{c[0]}\\}$

", "

$\\mathbb{R}$

"], "unitTests": [], "showCellAnswerState": true, "displayType": "radiogroup", "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "distractors": ["", ""], "displayColumns": "1", "type": "1_n_2", "useCustomName": false, "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "prompt": "

Which of the following is the domain of

\\[h(r)=\\simplify{(r-{c[0]})/(r-{c[0]})}?\\]

", "minMarks": 0, "shuffleChoices": false, "customMarkingAlgorithm": ""}], "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

Division is defined for all real numbers except zero. So when a function involves division (such as a rational function), any input that will result in a division by zero is not allowed. Rational functions are simply a fraction/division of polynomials, so the only real numbers that are not in the domain of a rational function are the roots of the polynomial in the denominator. Sometimes, (for example part c above) you will need to factorise the denominator to determine its roots.

d) Even though 'something divided by itself is 1' division by zero is still undefined. So the domain of $h$ is not all of $\\mathbb{R}$, the domain does not include the number $\\var{c[0]}$. In other words, $h(\\var{c[0]})$ is undefined but for all other $r$, $h(r)=1$.

", "rulesets": {}, "ungrouped_variables": ["out", "inp", "num", "n", "c", "rat", "a", "b", "set_of_holes"], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}