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Question
Let f = {(1, −1), (4, −2), (9, −3), (16, 4)} and g = {(−1, −2), (−2, −4), (−3, −6), (4, 8)}. Show that gof is defined while fog is not defined. Also, find gof.
Solution
f = {(1, −1), (4, −2), (9, −3), (16, 4)} and g = {(−1, −2), (−2, −4), (−3, −6), (4, 8)}
f : {1, 4, 9, 16} → {-1, -2, -3, 4} and g : {-1, -2, -3, 4} → {-2, -4, -6, 8}
Co-domain of f = domain of g
So, gof exists and gof : {1, 4, 9, 16} → {-2, -4, -6, 8}
(gof) (1) = g (f (1)) = g (−1) = −2
(gof) (4) = g (f (4))=g (−2) = −4
(gof) (9) = g (f (9)) = g (−3) = −6
(gof) (16) =g (f (16)) =g (4) = 8
So, gof = { (1, −2), (4, −4), (9, −6), (16, 8) }
But the co-domain of g is not same as the domain of f.
So, fog does not exist.
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