Card 1 of 20
EasyStudy card 1
Prompt
What does \((f\circ g)(x)\) mean?
Answer
\(f(g(x))\); apply \(g\) first, then \(f\).
MathChapter 12: Composition, Recursion, and Exponential Functions
Flashcards
Composition of Functions Flashcards
Twenty recall-focused cards on inner-first evaluation, order, representations, domain gates, and common composition traps.
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Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
How is \((f\circ g)(x)\) read aloud?
Answer
‘f of g of x.’
Card 3 of 20
EasyStudy card 3
Prompt
Which function is inner in \(f(g(x))\)?
Answer
\(g\), because it acts directly on \(x\).
Card 4 of 20
EasyStudy card 4
Prompt
Which function is outer in \(f(g(x))\)?
Answer
\(f\), because it receives \(g(x)\) as its input.
Card 5 of 20
EasyStudy card 5
Prompt
Must \(f\circ g\) equal \(g\circ f\)?
Answer
No. Function composition is generally order-dependent.
Card 6 of 20
EasyStudy card 6
Prompt
Does the circle in \(f\circ g\) mean multiplication?
Answer
No. It denotes composition, not the product \(f(x)g(x)\).
Card 7 of 20
EasyStudy card 7
Prompt
What is the inner-first workflow?
Answer
Find \(g(x)\), then use that entire result as the input of \(f\).
Card 8 of 20
MediumStudy card 8
Prompt
What is the first domain condition for \(f\circ g\)?
Answer
The original input \(x\) must be in the domain of \(g\).
Card 9 of 20
MediumStudy card 9
Prompt
What is the second domain condition for \(f\circ g\)?
Answer
The inner output \(g(x)\) must be in the domain of \(f\).
Card 10 of 20
MediumStudy card 10
Prompt
How do you compose from a table?
Answer
Look up the inner output, find that value in the input column, then read the outer output.
Card 11 of 20
MediumStudy card 11
Prompt
How do ordered pairs encode a function lookup?
Answer
In \((u,v)\), input \(u\) maps to output \(v\).
Card 12 of 20
MediumStudy card 12
Prompt
When is a table-based composition undefined?
Answer
When either the first input or the intermediate output has no required mapping.
Card 13 of 20
MediumStudy card 13
Prompt
Why use parentheses during symbolic composition?
Answer
They keep the entire inner expression together when it replaces each outer input.
Card 14 of 20
MediumStudy card 14
Prompt
For \(f(u)=u^2+u\), what must replace both occurrences of \(u\)?
Answer
The complete inner expression, such as \(g(x)\), in both places.
Card 15 of 20
MediumStudy card 15
Prompt
What does \(f(f(x))\) represent?
Answer
The output of one application of \(f\) used as the input of a second application.
Card 16 of 20
MediumStudy card 16
Prompt
What is a quick way to avoid reversing composition order?
Answer
Write the function nearest \(x\) as step 1 and the outer function as step 2.
Card 17 of 20
HardStudy card 17
Prompt
If \(g(2)=5\) and \(f(5)=-1\), find \(f(g(2))\).
Answer
\(-1\).
Card 18 of 20
HardStudy card 18
Prompt
If \(g(2)=5\) but \(5\) is outside the domain of \(f\), what is \(f(g(2))\)?
Answer
It is undefined.
Card 19 of 20
HardStudy card 19
Prompt
For \(f(x)=2x+1\) and \(g(x)=x-3\), find \(f(g(x))\).
Answer
\(2x-5\).
Card 20 of 20
HardStudy card 20
Prompt
For the same functions, find \(g(f(x))\).
Answer
\(2x-2\), confirming that the order changes the result.
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