Question

Let f = {(1, 2), (3, 5), (4, 1)) and g = {(2, 3), (5, 1), (1, 3)}. Then, gof = __________ and fog = __________.

Answer 1

Given: f = {(1, 2), (3, 5), (4, 1)) and g = {(2, 3), (5, 1), (1, 3)}

$$\begin{gathered} fog\left(x\right)=f\left(g\left(x\right)\right) \\ fog\left(1\right)=f\left(g\left(1\right)\right) \\ =f\left(3\right) \\ =5 \\ fog\left(2\right)=f\left(g\left(2\right)\right) \\ =f\left(3\right) \\ =5 \\ fog\left(5\right)=f\left(g\left(5\right)\right) \\ =f\left(1\right) \\ =2 \\ \mathrm{Hence},fog=\left\{\left(1,5\right),\left(2,5\right),\left(5,2\right)\right\} \\ gof\left(x\right)=g\left(f\left(x\right)\right) \\ gof\left(1\right)=g\left(f\left(1\right)\right) \\ =g\left(2\right) \\ =3 \\ gof\left(3\right)=g\left(f\left(3\right)\right) \\ =g\left(5\right) \\ =1 \\ gof\left(4\right)=g\left(f\left(4\right)\right) \\ =g\left(1\right) \\ =3 \\ \mathrm{Hence},gof=\left\{\left(1,3\right),\left(3,1\right),\left(4,3\right)\right\}. \end{gathered}$$

Hence, gof = $\underline{\left\{\left(1,3\right),\left(3,1\right),\left(4,3\right)\right\}}$ and fog = $\underline{\left\{\left(1,5\right),\left(2,5\right),\left(5,2\right)\right\}}$.