// Numbas version: exam_results_page_options
{"name": "Introduction to Functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Introduction to Functions", "tags": [], "metadata": {"description": "Introducing function notation, including evaluating a function for a given input.", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>You are given the following functions:</p>\n<p>$f(x) = \\var{a}x+\\var{b}$</p>\n<p>$g(x) = \\var{c}x-\\var{d}$&nbsp;</p>\n<p>$h(x) = \\var{d}x^2-\\var{b}x+\\var{c-1}$&nbsp;</p>", "advice": "<p><strong>(i)</strong> To find $f(\\var{x})$ we simply substitute $x=\\var{x}$ into our function $f(x) = \\var{a}x+\\var{b}$</p>\n<p>Hence&nbsp;$f(\\var{x})=\\var{a}(\\var{x})+\\var{b}=\\var{f(x)}$</p>\n<p></p>\n<p><strong>(ii)</strong> To find $h(\\var{x+2})$ we simply substitute $x=\\var{x+2}$ into our function $h(x) = \\var{d}x^2-\\var{b}x+\\var{c-1}$&nbsp;</p>\n<p>Hence&nbsp;$h(\\var{x+2})=\\var{d}(\\var{x+2})^2-\\var{b}(\\var{x+2})+\\var{c-1}=\\var{h(x+2)}$</p>\n<p></p>\n<p><strong>(iii)</strong> To find $g(x^2)$&nbsp;given that $x=\\var{x}$ we note that $x^2=\\var{x}^2=\\var{x^2}$</p>\n<p>Hence we just need to calculate $g(\\var{x^2})= \\var{c}(\\var{x^2})-\\var{d}=\\var{g(x^2)}$</p>\n<p></p>\n<p><strong>(iv)</strong> From (i) we note that&nbsp;$f(\\var{x})=\\var{f(x)}$ and hence $ff(\\var{x})=f(f(\\var{x}))=f(\\var{f(x)})=\\var{a}(\\var{f(x)})+\\var{b}=\\var{f(f(x))}$</p>\n<p></p>\n<p><strong>(v)</strong>&nbsp;We note that $gh(\\var{x+2})=g(h(\\var{x+2}))=g(\\var{h(x+2)})$ using our answer from (ii)</p>\n<p>But then $g(\\var{h(x+2)}) =&nbsp;\\var{c}(\\var{h(x+2)})-\\var{d}=\\var{g(h(x+2))}$&nbsp;</p>\n<p></p>\n<p></p>\n<p><strong>(vi)</strong>&nbsp;Note that $hg(\\var{x+2})=h(g(\\var{x+2}))$</p>\n<p>So first we must calculate $g(\\var{x+2})=\\var{c}(\\var{(x+2)})-\\var{d}=\\var{g(x+2)}$</p>\n<p>Then&nbsp;$hg(\\var{x+2}) = h(\\var{g(x+2)}) =&nbsp;\\var{d}(\\var{g(x+2)})^2-\\var{b}(\\var{g(x+2)})+\\var{c-1} =\\var{h(g(x+2))}$&nbsp;</p>\n<p></p>\n<p>Note the difference between (v) and (vi):</p>\n<p>To find $hg(\\var{x+2})$ we first find $g(\\var{x+2})$, then apply $h$ to this answer</p>\n<p>To find $gh(\\var{x+2})$ we first find $h(\\var{x+2})$, then apply $g$ to this answer</p>\n<p></p>\n<p><strong>(vii)</strong> $h(\\var{x+2})g(\\var{x+2})$ simply means&nbsp;$h(\\var{x+2}) \\times g(\\var{x+2})=\\var{h(x+2)} \\times \\var{g(x+2)}=\\var{h(x+2)*g(x+2)}$</p>", "rulesets": {}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2 .. 8#1)", "description": "", "templateType": "randrange"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1 .. 8#1)", "description": "", "templateType": "randrange"}, "c": {"name": "c", "group": "Ungrouped 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"scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Calculate:</p>\n<p>(i) $f(\\var{x})=$[[0]]</p>\n<p>(ii) $h(\\var{x+2})=$[[1]]</p>\n<p>(iii) $g(x^2)=$ [[2]] given that $x=\\var{x}$</p>\n<p>(iv) $ff(\\var{x})=$[[3]]</p>\n<p>(v) $gh(\\var{x+2})=$[[4]]</p>\n<p>(vi) $hg(\\var{x+2})=$[[5]]</p>\n<p>(vii) $h(\\var{x+2})g(\\var{x+2})=$[[6]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "gap 1", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "f({x})", "maxValue": "f({x})", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 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