3.3 Composite functions
Definition 3.11. Given functions \(f\) and \(g\), the composite \(f\circ g\) is defined by \[(f\circ g)(x)=f\big (g(x)\big ).\] Here \(g\) is the inside function and \(f\) the outside one.
Read it from the inside out: apply \(g\) first, then feed the answer into \(f\). The notation is the reverse of the order of work, which is what makes it easy to get backwards.
Example 3.12. Let \(f(x)=2x-1\) and \(g(x)=3x\). Find \[\text {(a) } (g\circ f)(x),\qquad \text {(b) } (f\circ g)(x), \qquad \text {(c) } (f\circ f)(x),\qquad \text {(d) } (g\circ g)(x).\]
Solution. (a) Inside first: \(f(x)=2x-1\). Feed that into \(g\), which triples its input: \[(g\circ f)(x)=g(2x-1)=3(2x-1)=6x-3.\]
(b) Now the other order. Inside is \(g(x)=3x\); feed it into \(f\), which doubles and subtracts \(1\): \[(f\circ g)(x)=f(3x)=2(3x)-1=6x-1.\]
(c) \[(f\circ f)(x)=f(2x-1)=2(2x-1)-1=4x-2-1=4x-3.\]
(d) \[(g\circ g)(x)=g(3x)=3(3x)=9x.\]
Note 3.13. Compare (a) and (b): \(6x-3\) against \(6x-1\). Composition is not commutative — \(f\circ g\) and \(g\circ f\) are usually different functions. Order matters, and it is not optional.
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