I proved the following two elementary but useful theorems:

Theorem For every funcoids $latex f$, $latex g$:

  1. If $latex \mathrm{im}\, f \supseteq \mathrm{im}\, g$ then $latex \mathrm{im}\, (g\circ f) = \mathrm{im}\, g$.
  2. If $latex \mathrm{im}\, f \subseteq \mathrm{im}\, g$ then $latex \mathrm{dom}\, (g\circ f) = \mathrm{dom}\, g$.

Theorem For every reloids $latex f$, $latex g$:

  1. If $latex \mathrm{im}\, f \supseteq \mathrm{im}\, g$ then $latex \mathrm{im}\, (g\circ f) = \mathrm{im}\, g$.
  2. If $latex \mathrm{im}\, f \subseteq \mathrm{im}\, g$ then $latex \mathrm{dom}\, (g\circ f) = \mathrm{dom}\, g$.

See this Web page and especially this online article.

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