Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by

The functions $f:\{1,3,4\} \rightarrow\{1,2,5\}$ and $g:\{1,2,5\} \rightarrow\{1,3\}$ are defined as

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

$\begin{array}{ll}g \circ f(1)=g(f(1))=g(2)=3 & {[f(1)=2 \text { and } g(2)=3]} \\ g \circ f(3)=g(f(3))=g(5)=1 & {[f(3)=5 \text { and } g(5)=1]} \\ g \circ f(4)=g(f(4))=g(1)=3 & {[f(4)=1 \text { and } g(1)=3]} \\ \therefore g \circ f=\{(1,3),(3,1),(4,3)\} & \end{array}$