Question

If the mapping f and g are given by
f = {(1, 2), (3, 5), (4, 1)}
and g = {(2, 3), (5, 1), (1, 3)},
write down pairs in the mapping fog and gof.

Answer 1

Domain f = {1,3,4}, Range f = {2, 5, 1}
Domain g = {2,5,1}, Range g = {1,3}
`because` Range f = Dom g = {(2,5,1)}
`therefore` gof mapping is defined.
Then, gof mapping defined following way

We see that, f(1) = 2, f(3) = 5, f(4) = 1
and g(2) = 3, g(5) = 1, g(1) = 3
`therefore` (gof)(1) = g{f(1)} = g(2) = 3
(gof)(3) = g{f(3)} = g(5) = 1
(gof)(4) = g{f(4)} = g(1) = 3
Hence, gof = {(1, 3), (3, 1), (4, 3)}
Now, since Range of f `sub` Dom f
`therefore` fog is defined
Then, fog mapping defined following way

We see that, g(2) = 3, g(5) = 1, g(1) = 3
f(1) = 2, f(3) = 5, f(4) = 1
`therefore` (fog)(2) = f {g(2)} = f(3)=5
(fog)(5) = f{g(5)} = f(1)=2
(fog)(1) = f{g(1)} = f(3) = 5
Hence, fog = {(2,5),(5,2),(1,5)}