Find the inverse of a composite function by finding the inverses of two functions and then the composite of these; and by finding the composite of two functions then finding the inverse. The question then concludes by asking students to compare their two answers and verify they're equivalent.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\nGive all of your answers in terms of $x$.
$f^{-1}(x)$ is the function with the property that $f^{-1}(f(x)) = x$.
\nTo find this, we first set $x=f(y)$ and rearrange to find $y$ in terms of $x$, i.e. $y = f^{-1}(x)$.
\n\\begin{align}
f(y)=\\simplify{{a[0]}y+{a[1]}}&=x\\\\
\\simplify{{a[0]}y}&=x-\\var{a[1]}\\\\[1em]
y&=\\simplify[]{(x-{a[1]})/{a[0]}}\\\\[1em]
f^{-1}(x)&=\\simplify{(x-{a[1]})/{a[0]}}\\text{.}\\\\
\\end{align}
We use the same method as part a) to find $g^{-1}(x)$:
\n\\begin{align}
g(y)=\\simplify{{a[3]}y-{a[2]}}&=x\\\\
\\simplify{{a[3]}y}&=x+\\var{a[2]}\\\\[1em]
y&=\\simplify[]{(x+{a[2]})/{a[3]}}\\\\[1em]
g^{-1}(x)&=\\simplify{(x+{a[2]})/{a[3]}}\\text{.}\\\\
\\end{align}
$(g^{-1} \\circ f^{-1})(x)$ is the function which first applies $f^{-1}(x)$ and then applies $g^{-1}$ to the result of that.
\nWe use the previous answers: $f^{-1}(x)=\\simplify{(x-{a[1]})/{a[0]}}$ and $g^{-1}(x)=\\simplify{(x+{a[2]})/{a[3]}}$ to find the definition of $(g^{-1} \\circ f^{-1})(x)$ by substituting $f^{-1}(x)$ everywhere $x$ occurs in the definition of $g^{-1}(x)$.
\n\\begin{align}
(g^{-1}\\circ f^{-1}) (x)&=g^{-1}(f^{-1}(x))\\\\[1em]
&=g^{-1} \\left( \\simplify[]{(x-{a[1]})/{a[0]}} \\right) \\\\[1em]
&=\\frac{\\left(\\simplify[]{(x-{a[1]})/{a[0]}}\\right)+\\simplify[]{{a[2]}}}{\\var{a[3]}\\text{.}}
\\end{align}
$(f \\circ g)(x)$ is the function which first applies $g(x)$ and then applies $f$ to the result of that.
\nWe find the definition of $(f \\circ g)(x)$ by substituting $g(x)$ everywhere that $x$ occurs in the definition of $f(x)$.
\n\\begin{align}
(f\\circ g)(x)&=f(g(x))\\\\
&=f(\\simplify{{a[3]}x-{a[2]}})\\\\
&=\\simplify{{a[0]}({a[3]}x-{a[2]})+{a[1]}}
\\end{align}
Now that we have the definition of $(f \\circ g)(x)$, we can find its inverse by using the same method as in parts a) and b).
\n\\begin{align}
(f \\circ g)(y) &= x \\\\
\\simplify{{a[0]}({a[3]}y-{a[2]})+{a[1]}}&=x\\\\
\\simplify{{a[0]}({a[3]}y-{a[2]})}&=x-\\var{a[1]}\\\\[1em]
\\simplify{{a[3]}y-{a[2]}}&=\\frac{(x-\\var{a[1]})}{\\var{a[0]}}\\\\[1em]
\\simplify{{a[3]}y}&=\\left( \\frac{(x-\\var{a[1]})}{\\var{a[0]}} \\right) +\\var{a[2]}\\\\[1em]
y&=\\frac{\\left(\\frac{(x-\\var{a[1]})}{\\var{a[0]}}\\right)+\\var{a[2]})}{\\var{a[3]}}\\\\[1em]
(f\\circ g)^{-1}(x)&=\\frac{\\left(\\frac{(x-\\var{a[1]})}{\\var{a[0]}}\\right)+\\var{a[2]}}{\\var{a[3]}}\\\\[1em]
\\end{align}
We can see that in this case $(f\\circ g)^{-1}(x) = (g^{-1}\\circ f^{-1}) (x)$.
\nSo long as the inverses of $f$ and $g$ exist and they can be composed, it is always true that \\[(f \\circ g)^{-1}(x) \\equiv (g^{-1} \\circ f^{-1}) (x)\\text{.}\\]
", "rulesets": {}, "extensions": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "repeat(random(5..9),5)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "repeat(random(2..5),5)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Find $f^{-1}(x)$ when $\\simplify{f(x)={a[0]}x+{a[1]} }$.
\n$\\displaystyle f^{-1}(x)=$ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(x-{a[1]})/{a[0]}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Find $g^{-1}(x)$ when $\\simplify{g(x)={a[3]}x-{a[2]} }$.
\n$\\displaystyle g^{-1}(x)=$ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(x+{a[2]})/{a[3]}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Use your above answers for $f^{-1}(x)$ and $g^{-1}(x)$ to find the inverse, composed function, $(g^{-1}\\circ f^{-1}) (x)$ terms of $x$:
\n
$\\displaystyle (g^{-1}\\circ f^{-1}) (x)$ =[[0]]
Using:
\\[
\\begin{align}
f(x)&=\\simplify{{a[0]}x+{a[1]} }\\\\
&\\text{ and } \\\\
g(x)&=\\simplify{{a[3]}x-{a[2]} }\\text{,}
\\end{align}
\\]
find $(f\\circ g)(x)$, the composition of $f(x)$ with $g(x)$.
$\\displaystyle (f\\circ g)(x)=$ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{a[0]}({a[3]}x-{a[2]})+{a[1]}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Use your above answer for $(f\\circ g)(x)$ to find the inverse, composed function,$(f\\circ g)^{-1}(x)$ in terms of $x$:
\n$\\displaystyle (f\\circ g)^{-1}(x)=$ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(x-{a[1]}+({a[2]}{a[0]}))/({a[3]}*{a[0]})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "When should your answer for c), $(g^{-1}\\circ f^{-1}) (x)$ be the same as your answer for e) $(f\\circ g)^{-1}(x)$?
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", "Only when $f(x)=g(x)$
", "Only when $f(x)$ is in the same family as $g(x)$
", "Always, provided that composite and inverse functions exist
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