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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.
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Solution
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}
(gof) (3)=g (f (3))=g (1) =3
(gof) (9)=g (f (9))=g (3)=3
(gof) (12)=g (f (12))=g (4)=9
⇒ gof ={(3, 3), (9, 3), (12, 9)}
Co-domain of g is a subset of the domain of f.
So, fog exists and fog : {1, 3, 4, 5} → {3, 9, 12}
(fog) (1)=f (g (1))=f (3)=1
(fog) (3)=f (g (3))=f (3)=1
(fog) (4)=f (g (4))=f (9)=3
(fog) (5)=f (g (5))=f (9)=3
⇒ fog={(1, 1), (3, 1), (4, 3), (5, 3)}
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