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It is a fact that any old function that you might pick factors into a functional composition of two other functions, a surjective ("onto") function and an injective ("one-to-one") function, in the present example pictured below:

It is a fact that any old function that you might pick factors into a composition of two other functions, a surjective ("onto") function and an injective ("one-to-one") function, in the present example pictured below:

{| align="center" cellpadding="10" style="text-align:center; width:90%"

{| align="center" cellpadding="8" style="text-align:center"

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| [[Image:Factorization Function Example 2.jpg|500px]]

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o---------------------------------------o

| <math>\text{Figure 2.  Factorization}~ f = g \circ h</math>

|                                       |

|  Source X  =  {1, 2, 3, 4,    5}     |

|      g   |      \ | /    \  /      |

|          v        \|/      \ /        |

|      h   |        |        |        |

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X & = & \{ 1, 2, 3, 4, 5 \}

X & = & \{ 1, 2, 3, 4, 5 \}

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M & = & \{ Q, T \}

M & = & \{ m, n \}

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Y & = & \{ p, q, r, s, t, u \}

Y & = & \{ p, q, r, s, t, u \}