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That is, the value of <math>g \circ f</math> is obtained by first applying {{math|''f''}} to {{math|''x''}} to obtain {{math|1=''y'' = ''f''(''x'')}} and then applying {{math|''g''}} to the result {{mvar|y}} to obtain {{math|1=''g''(''y'') = ''g''(''f''(''x''))}}. In this notation, the function that is applied first is always written on the right.
The composition <math>g\circ f</math> is an [[operation (mathematics)|operation]] on functions that is defined only if the codomain of the first function is the ___domain of the second one. Even when both <math>g \circ f</math> and <math>f \circ g</math> satisfy these conditions, the composition is not necessarily [[commutative property|commutative]], that is, the functions <math>g \circ f</math> and <math> f \circ g</math> need not be equal,
The function composition is [[associative property|associative]] in the sense that, if one of <math>(h\circ g)\circ f</math> and <math>h\circ (g\circ f)</math> is defined, then the other is also defined, and they are equal, that is, <math>(h\circ g)\circ f = h\circ (g\circ f).</math> Therefore, it is usual to just write <math>h\circ g\circ f.</math>
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