Give an example of a function which represents all types of a function. Find the composite function (f o g) (x) given that f = {(1,6), (4,7), (5,0)} and g = {(6,1), (7,4), (0,5)}

Let us give an example of a function which represents all types of a function. Consider the function $h=\{(a,0),(b,1)\}.$ Since for $a\ne b$ we have that $f(a)=0\ne 1=f(b),$ we conclude that this function is injection. Taking into account that $f^{-1}(0)=\{a\}\ne \emptyset, \ f^{-1}(1)=\{b\}\ne \emptyset,$ we conclude that the function $f$ is surjection, and hence this function is bijection.

Let us find the composite function $(f\circ g) (x)$ given that $f = \{(1,6), (4,7), (5,0)\}$ and $g = \{(6,1), (7,4), (0,5)\}$:

(f\circ g) (0)=f(g(0))=f(5)=0,\ (f\circ g) (6)=f(g(6))=f(1)=6,\

$(f\circ g) (7)=f(g(7))=f(4)=7.$

Therefore, $f\circ g = \{(0,0), (6,6), (7,7)\}.$