$f=\left\{ (1\,,\,2)\,,\,(2\,,\,3)\,,\,(-1\,,\,0)\,,\,(0\,,\,2) \right\}\Rightarrow {{D}_{f}}=\left\{ -1\,,\,0\,,\,1\,,\,2 \right\}$

$g=\left\{ (1\,,\,a)\,,\,(b\,,\,1)\,,\,(0\,,\,2)\,,\,(2\,,\,{{a}^{2}}) \right\}\Rightarrow {{D}_{g}}=\left\{ 1\,,\,0\,,\,2\,,\,b \right\}$

${{D}_{fog}}=\left\{ x\left| x\in {{D}_{g}}\,,\,g(x)\in {{D}_{f}} \right. \right\}=\left\{ -1\,,\,0\,,\,1 \right\}$

$1\in {{D}_{fog}}\Rightarrow g(1)\in {{D}_{f}}\Rightarrow a\in {{D}_{f}}\Rightarrow a\in {{D}_{f}}\Rightarrow a\in \left\{ 1\,,\,2\,,\,-1\,,\,0 \right\}$

$(-1)\in {{D}_{fog}}\Rightarrow (-1)\in {{D}_{g}}\,,\,g(-1)\in {{D}_{f}}\Rightarrow b=-1\,,\,g(-1)=g(b)=1\in {{D}_{f}}$

اما چون $2\notin {{D}_{fog}}$ و $2\in {{D}_{g}}$، پس باید $g(2)\notin {{D}_{f}}$، بنابراین:

$g(2)\notin {{D}_{f}}\Rightarrow {{a}^{2}}\notin {{D}_{f}}\xrightarrow{a\in {{D}_{f}}}a=2$