// Numbas version: exam_results_page_options
{"name": "Evaluating Composite Functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Evaluating Composite Functions", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>This question will be about function composition. You will be given two functions, along with an input, before being asked to evaluate the composite function at this input.</p>\n<p>The first function is $f \\colon \\var{input_set} \\rightarrow \\mathbb{Z}$, defined by {input_function}.</p>\n<p>The second function is $g \\colon \\mathbb{Z} \\rightarrow \\var{output_set}$, defined by {output_function}.</p>", "advice": "<p>QQQ Add link to function composition example from the online notes.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"input_A1": {"name": "input_A1", "group": "Ungrouped variables", "definition": "random(-100 .. 100#1)", "description": "<p>A random integer</p>", "templateType": "randrange", "can_override": false}, "input_A2": {"name": "input_A2", "group": "Ungrouped variables", "definition": "random(0 .. 100#1)", "description": "<p>A random natural number</p>", "templateType": "randrange", "can_override": false}, "input_A3": {"name": "input_A3", "group": "Ungrouped variables", "definition": "random(-50 .. 50#0)", "description": "<p>A random \"real\" number</p>", "templateType": "randrange", "can_override": false}, "input_A4": {"name": "input_A4", "group": "Ungrouped variables", "definition": "random(\"a\",\"b\",\"c\")", "description": "<p>A random element of the set {a,b,c}</p>", "templateType": "anything", "can_override": false}, "output_f1": {"name": "output_f1", "group": "Ungrouped variables", "definition": "input_A1", "description": "", "templateType": "anything", "can_override": false}, "output_f2": {"name": "output_f2", "group": "Ungrouped variables", "definition": "-input_A2", "description": "", "templateType": "anything", "can_override": false}, "output_f3": {"name": "output_f3", "group": "Ungrouped variables", "definition": "floor(input_A3)", "description": "", "templateType": "anything", "can_override": false}, "output_f4": {"name": "output_f4", "group": "Ungrouped variables", "definition": "if(input_A4=\"a\",4,if(input_A4=\"b\",27,-3))", "description": "", "templateType": "anything", "can_override": false}, "output_g1f1": {"name": "output_g1f1", "group": "Ungrouped variables", "definition": "output_f1", "description": "", "templateType": "anything", "can_override": false}, "output_g2f1": {"name": "output_g2f1", "group": "Ungrouped variables", "definition": "mod(output_f1,4)", "description": "", "templateType": "anything", "can_override": false}, "output_g3f1": {"name": "output_g3f1", "group": "Ungrouped variables", "definition": "if(mod(output_f1,2)=0,\"Blue\",\"Yellow\")", "description": "", "templateType": "anything", "can_override": false}, "output_g1f2": {"name": "output_g1f2", "group": "Ungrouped variables", "definition": "output_f2", "description": "", "templateType": "anything", "can_override": false}, "output_g2f2": {"name": "output_g2f2", "group": "Ungrouped variables", "definition": "mod(output_f2,4)", "description": "", "templateType": "anything", "can_override": false}, "output_g3f2": {"name": "output_g3f2", "group": "Ungrouped variables", "definition": "if(mod(output_f2,2)=0,\"Blue\",\"Yellow\")", "description": "", "templateType": "anything", "can_override": false}, "output_g1f3": {"name": "output_g1f3", "group": "Ungrouped variables", "definition": "output_f3", "description": "", "templateType": "anything", "can_override": false}, "output_g2f3": {"name": "output_g2f3", "group": "Ungrouped variables", "definition": "mod(output_f3,4)", "description": "", "templateType": "anything", "can_override": false}, "output_g3f3": {"name": "output_g3f3", "group": "Ungrouped variables", "definition": "if(mod(output_f3,2)=0,\"Blue\",\"Yellow\")", "description": "", "templateType": "anything", "can_override": false}, "output_g1f4": {"name": "output_g1f4", "group": "Ungrouped variables", "definition": "output_f4", "description": "", "templateType": "anything", "can_override": false}, "output_g2f4": {"name": "output_g2f4", "group": "Ungrouped variables", "definition": "mod(output_f4,4)", "description": "", "templateType": "anything", "can_override": false}, "output_g3f4": {"name": "output_g3f4", "group": "Ungrouped variables", "definition": "if(mod(output_f4,2)=0,\"Blue\",\"Yellow\")", "description": "", "templateType": "anything", "can_override": false}, "A1": {"name": "A1", "group": "Ungrouped variables", "definition": "safe(\"$\\\\mathbb{Z}$\")", "description": "", "templateType": "long plain string", "can_override": false}, "A2": {"name": "A2", "group": "Ungrouped variables", "definition": "safe(\"$\\\\mathbb{N}$\")", "description": "", "templateType": "long plain string", "can_override": false}, "A3": {"name": "A3", "group": "Ungrouped variables", "definition": "safe(\"$\\\\mathbb{R}$\")", "description": "", "templateType": "long plain string", "can_override": false}, "A4": {"name": "A4", "group": "Ungrouped variables", "definition": "safe(\"$\\{a,b,c\\}$\")", "description": "", "templateType": "long plain string", "can_override": false}, "input_set": {"name": "input_set", "group": "Ungrouped variables", "definition": "random(\"{A1}\",\"{A2}\",\"{A3}\",\"{A4}\")", "description": "", "templateType": "anything", "can_override": false}, "C1": {"name": "C1", "group": "Ungrouped variables", "definition": "safe(\"$\\\\mathbb{Z}$\")", "description": "", "templateType": "long plain string", "can_override": false}, "C2": {"name": "C2", "group": "Ungrouped variables", "definition": "safe(\"$\\\\mathbb{Z}_4$\")", "description": "", "templateType": "long plain string", "can_override": false}, "C3": {"name": "C3", "group": "Ungrouped variables", "definition": "safe(\"$\\{\\\\text{Blue},\\\\text{Yellow}\\}$\")", "description": "", "templateType": "long plain string", "can_override": false}, "output_set": {"name": "output_set", "group": "Ungrouped variables", "definition": "random(\"{C1}\",\"{C2}\",\"{C3}\")", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "safe(\"$f(n) = n$\")", "description": "", "templateType": "long plain string", "can_override": false}, "f2": {"name": "f2", "group": "Ungrouped variables", "definition": "safe(\"$f(n) = -n$\")", "description": "", "templateType": "long plain string", "can_override": false}, "f3": {"name": "f3", "group": "Ungrouped variables", "definition": "safe(\"$f(x)$ is equal to $x$ rounded down to the nearest integer. E.g., $f(-3.5) = -4$.\")", "description": "", "templateType": "long plain string", "can_override": false}, "f4": {"name": "f4", "group": "Ungrouped variables", "definition": "safe(\"$f(a) = 4$, $f(b) = 27$, and $f(c) = -3$\")", "description": "", "templateType": "long plain string", "can_override": false}, "input_index": {"name": "input_index", "group": "Ungrouped variables", "definition": "random(1,2,3,4)", "description": "", "templateType": "anything", "can_override": false}, "input_function": {"name": "input_function", "group": "Ungrouped variables", "definition": "if(input_set=A1,f1,if(input_set=A2,f2,if(input_set=A3,f3,f4)))", "description": "", "templateType": "anything", "can_override": false}, "g1": {"name": "g1", "group": "Ungrouped variables", "definition": "safe(\"$g(m) = m$\")", "description": "", "templateType": "long plain string", "can_override": false}, "g2": {"name": "g2", "group": "Ungrouped variables", "definition": "safe(\"$g(m) = m \\\\operatorname{mod} 4$\")", "description": "", "templateType": "long plain string", "can_override": false}, "g3": {"name": "g3", "group": "Ungrouped variables", "definition": "safe(\"$g(m) = \\\\text{Blue, if } x = 0 \\\\operatorname{mod} 2$ or $g(m) = \\\\text{Yellow, if } x = 1 \\\\operatorname{mod} 2$\")", "description": "", "templateType": "long plain string", "can_override": false}, "output_function": {"name": "output_function", "group": "Ungrouped variables", "definition": "if(output_set=C1,g1,if(output_set=C2,g2,g3))", "description": "", "templateType": "anything", "can_override": false}, "input_value": {"name": "input_value", "group": "Ungrouped variables", "definition": "if(input_set=A1,input_A1,if(input_set=A2,input_A2,if(input_set=A3,input_A3,input_A4)))", "description": "", "templateType": "anything", "can_override": false}, "output_value": {"name": "output_value", "group": "Ungrouped variables", "definition": "if(input_set=A1 and output_set=C1,output_g1f1,\n  if(input_set=A2 and output_set=C1,output_g1f2,\n  if(input_set=A3 and output_set=C1,output_g1f3,\n  if(input_set=A4 and output_set=C1,output_g1f4,\n  if(input_set=A1 and output_set=C2,output_g2f1,\n  if(input_set=A2 and output_set=C2,output_g2f2,\n  if(input_set=A3 and output_set=C2,output_g2f3,\n  if(input_set=A4 and output_set=C2,output_g2f4,\n  if(input_set=A1 and output_set=C3,output_g3f1,\n  if(input_set=A2 and output_set=C3,output_g3f2,\n  if(input_set=A3 and output_set=C3,output_g3f3,\n  output_g3f4)))))))))))\n", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["input_A1", "input_A2", "input_A3", "input_A4", "output_f1", "output_f2", "output_f3", "output_f4", "output_g1f1", "output_g2f1", "output_g3f1", "output_g1f2", "output_g2f2", "output_g3f2", "output_g1f3", "output_g2f3", "output_g3f3", "output_g1f4", "output_g2f4", "output_g3f4", "A1", "A2", "A3", "A4", "input_set", "C1", "C2", "C3", "output_set", "f1", "f2", "f3", "f4", "input_index", "input_function", "g1", "g2", "g3", "output_function", "input_value", "output_value"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "patternmatch", "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>You are given input value $m = \\var{input_value} \\in \\var{input_set}$. What is the value of $(g \\circ f)(m)$?</p>", "answer": "{output_value}", "displayAnswer": "", "matchMode": "exact"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Alexander Corner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5328/"}]}]}], "contributors": [{"name": "Alexander Corner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5328/"}]}