Question

Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.

Answer 1

f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}

f : {3, 9, 12} → {1, 3,4} and g : {1, 3, 4, 5} → {3, 9}

Co-domain of f is a subset of the domain of g.
So, gof exists and gof : {3, 9, 12} → {3, 9}
$$\begin{gathered} \left(gof\right)\left(3\right)=g\left(f\left(3\right)\right)=g\left(1\right)=3 \\ \left(gof\right)\left(9\right)=g\left(f\left(9\right)\right)=g\left(3\right)=3 \\ \left(gof\right)\left(12\right)=g\left(f\left(12\right)\right)=g\left(4\right)=9 \\ \Rightarrow gof=\left\{\left(3,3\right),\left(9,3\right),\left(12,9\right)\right\} \end{gathered}$$
Co-domain of g is a subset of the domain of f.
So, fog exists and fog : {1, 3, 4, 5} → {3, 9, 12}
$$\begin{gathered} \left(fog\right)\left(1\right)=f\left(g\left(1\right)\right)=f\left(3\right)=1 \\ \left(fog\right)\left(3\right)=f\left(g\left(3\right)\right)=f\left(3\right)=1 \\ \left(fog\right)\left(4\right)=f\left(g\left(4\right)\right)=f\left(9\right)=3 \\ \left(fog\right)\left(5\right)=f\left(g\left(5\right)\right)=f\left(9\right)=3 \\ \Rightarrow fog=\left\{\left(1,1\right),\left(3,1\right),\left(4,3\right),\left(5,3\right)\right\} \end{gathered}$$