$f(x) = \\simplify{{mf}x+{cf}}$.
\n$g(x) = \\simplify{{mg}x+{cg}}$.
\n$h(x) = \\simplify{{mh}x+{ch}}$.
\n\n\n$f(\\var{x[1]}) =$ [[0]]
\n$h(\\var{x[2]}) = $ [[1]]
\n$f(g(\\var{x[3]}))= $ [[2]]
\n$g(h(\\var{x[4]})) = $ [[3]]
\n$g(g(\\var{x[5]})) = $ [[4]]
\n$f^{-1}(\\var{fx[6]}) = $ [[5]]
\n$h^{-1}(\\var{hx[7]}) = $ [[6]]
\n$g^{-1}(\\var{gx[8]}) = $ [[7]]
\n$h^{-1}(\\var{hx[9]}) = $ [[8]]
\n$g(g(\\var{x[10]})) = $ [[9]]
\nIf $a$ is any number, what is $g(g(a))$?
\n$g(g(a)) = $ [[10]]
\n\nIf $p$ is some number, what is $f^{-1}(p)$?
\n$f^{-1}(p) = $ [[11]]
\n\nIf $s$ is some number, what is $g^{-1}(s)$?
\n$g^{-1}(s) = $ [[12]]
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"extendBaseMarkingAlgorithm": true, "maxValue": "x[7]", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "marks": 1, "showFeedbackIcon": true, "type": "numberentry", "showCorrectAnswer": true, "scripts": {}, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "unitTests": [], "variableReplacementStrategy": "originalfirst"}, {"minValue": "x[8]", "allowFractions": false, "extendBaseMarkingAlgorithm": true, "maxValue": "x[8]", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "marks": 1, "showFeedbackIcon": true, "type": "numberentry", "showCorrectAnswer": true, "scripts": {}, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "unitTests": [], "variableReplacementStrategy": "originalfirst"}, {"minValue": "x[9]", "allowFractions": false, "extendBaseMarkingAlgorithm": true, "maxValue": "x[9]", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "marks": 1, "showFeedbackIcon": true, "type": "numberentry", "showCorrectAnswer": true, "scripts": {}, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "unitTests": [], "variableReplacementStrategy": "originalfirst"}, {"minValue": "ggx[10]", "allowFractions": false, "extendBaseMarkingAlgorithm": true, "maxValue": "ggx[10]", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "marks": 1, "showFeedbackIcon": true, "type": "numberentry", "showCorrectAnswer": true, "scripts": {}, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "unitTests": [], "variableReplacementStrategy": "originalfirst"}, {"checkingAccuracy": 0.001, "failureRate": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "expectedVariableNames": [], "checkVariableNames": false, "showPreview": true, "vsetRange": [0, 1], "marks": 1, "showFeedbackIcon": true, "answer": "a+{cg*2}", "showCorrectAnswer": true, "scripts": {}, "answerSimplification": "all", "type": "jme", "customMarkingAlgorithm": "", "variableReplacements": [], "vsetRangePoints": 5, "unitTests": [], "variableReplacementStrategy": "originalfirst"}, {"checkingAccuracy": 0.001, "failureRate": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "expectedVariableNames": [], "checkVariableNames": false, "showPreview": true, "vsetRange": [0, 1], "marks": 1, "showFeedbackIcon": true, "answer": "p/{mf}", "showCorrectAnswer": true, "scripts": {}, "answerSimplification": "all", "type": "jme", "customMarkingAlgorithm": "", "variableReplacements": [], "vsetRangePoints": 5, "unitTests": [], "variableReplacementStrategy": "originalfirst"}, {"checkingAccuracy": 0.001, "failureRate": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "expectedVariableNames": [], "checkVariableNames": false, "showPreview": true, "vsetRange": [0, 1], "marks": 1, "showFeedbackIcon": true, "answer": "s-{cg}", "showCorrectAnswer": true, "scripts": {}, "answerSimplification": "all", "type": "jme", "customMarkingAlgorithm": "", "variableReplacements": [], "vsetRangePoints": 5, "unitTests": [], "variableReplacementStrategy": "originalfirst"}], "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "sortAnswers": false, "customMarkingAlgorithm": "", "variableReplacements": [], "unitTests": [], "variableReplacementStrategy": "originalfirst"}], "preamble": {"js": "", "css": ""}, "advice": "See Lecture 4.1 and Workshop 4.2
", "ungrouped_variables": [], "functions": {"eqnline": {"definition": "// This functions plots a cubic with a certain number of\n// stationary points and roots.\n// It creates the board, sets it up, then returns an\n// HTML div tag containing the board.\n\n\n// Max and min x and y values for the axis.\nvar x_min = -6;\nvar x_max = 6;\nvar y_min = -20;\nvar y_max = 20;\n\n\n// First, make the JSXGraph board.\nvar div = Numbas.extensions.jsxgraph.makeBoard(\n '500px',\n '600px',\n {\n boundingBox: [x_min,y_max,x_max,y_min],\n axis: false,\n showNavigation: true,\n grid: true\n }\n);\n\n\n\n\n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar board = div.board; \n\n// create the x-axis and y-axis\nvar xaxis = board.create('axis',[[0,0],[1,0]]);\n\n// create the y-axis\nvar yaxis = board.create('axis',[[0,0],[0,1]], );\n\n\n\n\n// Plot the function.\n board.create('functiongraph',\n [function(x){ return m*x+c},x_min,x_max]);\n\n\n\nreturn div;", "parameters": [["m", "number"], ["c", "number"]], "language": "javascript", "type": "html"}}, "name": "Algebra: Functions, inverses and compositions [L3 Randomised]", "variable_groups": [{"variables": ["mf", "cf", "mg", "cg", "mh", "ch", "x", "fx", "gx", "hx", "fgx", "ghx", "ggx"], "name": "part a"}], "metadata": {"description": "A few simple functions are provided of the form ax, x+b and cx+d. Values of the functions, inverses and compositions are asked for. Most are numerical but the last few questions are algebraic.
", "licence": "Creative Commons Attribution 4.0 International"}, "extensions": ["jsxgraph"], "variables": {"cg": {"definition": "random(-5..5 except 0)", "description": "", "group": "part a", "name": "cg", "templateType": "anything"}, "ch": {"definition": "random(-5..5 except 0)", "description": "", "group": "part a", "name": "ch", "templateType": "anything"}, "gx": {"definition": "x*mg+vector(cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg)", "description": "", "group": "part a", "name": "gx", "templateType": "anything"}, "hx": {"definition": "x*mh+vector(ch,ch,ch,ch,ch,ch,ch,ch,ch,ch,ch,ch,ch,ch,ch,ch,ch,ch,ch,ch)", "description": "", "group": "part a", "name": "hx", "templateType": "anything"}, "mh": {"definition": "random(2..6)", "description": "", "group": "part a", "name": "mh", "templateType": "anything"}, "ggx": {"definition": "gx*mg+vector(cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg)", "description": "", "group": "part a", "name": "ggx", "templateType": "anything"}, "fgx": {"definition": "gx*mf+vector(cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf)", "description": "", "group": "part a", "name": "fgx", "templateType": "anything"}, "fx": {"definition": "x*mf+vector(cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf,cf)", "description": "", "group": "part a", "name": "fx", "templateType": "anything"}, "mg": {"definition": "1", "description": "", "group": "part a", "name": "mg", "templateType": "anything"}, "cf": {"definition": "0", "description": "", "group": "part a", "name": "cf", "templateType": "anything"}, "x": {"definition": "vector(shuffle(-4..4)+shuffle(-4..4)+shuffle(1..2))", "description": "", "group": "part a", "name": "x", "templateType": "anything"}, "ghx": {"definition": "hx*mg+vector(cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg,cg)", "description": "", "group": "part a", "name": "ghx", "templateType": "anything"}, "mf": {"definition": "random(-4..4 except [0,1])", "description": "", "group": "part a", "name": "mf", "templateType": "anything"}}, "tags": ["L3", "l3"], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "This is a non-calculator question.
", "type": "question", "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Abbi Mullins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2466/"}, {"name": "Matthew James Sykes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2582/"}]}]}], "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Abbi Mullins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2466/"}, {"name": "Matthew James Sykes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2582/"}]}