FoundationsConcept reference

Function composition

A composition applies an inner function to an input and then applies an outer function to the result.

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01Let g(x)=x+2 and f(x)=2x#

Let $g(x)=x+2$ and $f(x)=2x$ . In $f(g(3))$ , the inner rule $g$ receives 3 first. Its output then becomes the input to $f$ .

Let $g(x)=x+2$ and $f(x)=2x$ . First, $g(3)=5$ . Then $f(5)=10$ . The notation $(f\circ g)(3)$ means the same thing as $f(g(3))$ .

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If $g(x)=x+2$ and $f(x)=2x$ , what is $f(g(4))$ ?

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First $g(4)=6$ . Then $f(6)=12$ .

Order matters. Let $g(x)=x+2$ and $f(x)=2x$ . Then $g(f(3))$ doubles first, then adds 2: $g(6)=8$ . In the reverse order, $f(g(3))=10$ .

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In $g(f(5))$ , which rule receives 5 first?

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$f$

Compute the inner $f(5)$ first.

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A data pipeline first subtracts 1 from a reading, then triples that result. What final output comes from input 4?

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Stage one gives $4-1=3$ . Stage two gives $3\times3=9$ .

Key takeaway