San José State University
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The Associativity of
Function Composition

Let f( ) and g( ) be two functions. The composition of f and g, f°g, is f(g( )).

Let f, g and h be three functions and F, G and H are their domains of definition, respectively.

Consider f(g(h( ))). Associtivity requires that

(f°g), h( )) = (f(g°h( ))

Now consider a sequence of mappings. Suppose h maps x to y and g maps y to z whic f maps to w. The composite of g and h will map x directly to z, which is the mapped to w. Now conside the composition of f and g. After h maps x to y the composite of f and g then maps y directly to w. Thus both (f°g(h(x)) and f(g°h(x)) are equal w. Hence function composition is associative.

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