Let $f:\{ 1,\;3,\;4\} \to \{ 1,\;2,\;5\}$ and $g:\{ 1,\;2,\;5\} \to \{ 1,\;3]$ be given by $f = \{ (1,\;2),\;(3,\;5),\;(4,\;1)\}$ and $g = \{ (1,\;3),\;(2,\;3),\;(5,\;1)\} .$ Write down gof.

$f = \{ (1,\;2),\;(3,\;5),\;(4,\;1)\}$ and $g = \{ (1,\;3),\;(2,\;3),\;(5,\;1)\}$

Now, $f(1) = 2,\;f(3) = 5,\;f(4) = 1\;\;and\;\;g(1) = 3,\;g(2) = 3,\;\;g(5) = 1$
$(gof)(x) = g[f(x)] \Rightarrow g[f(1)] = g(2) = 3$

$g[f(3)] = g(5) = 1,\;\;g[f(4)] = g(1) = 3$

Hence, $gof = \{ (1,\;3),\;(3,\;1),\;(4,\;3)\} .$