Question

Let f = {(1, 2), (3, 4), (2, 2)} and g = {(2, 1), (3, 1), (4, 2)}. Find g ◦ f and f ◦ g.​

Answer 1

Answer :

gof = { (1,1) , (2,1) , (3,2) }

fog = { (2,2) , (3,2) , (4,2) }

Solution :

• Given : f = { (1,2) , (3,4) , (2,2) }

g = { (2,1) , (3,1) , (4,2) }

• To find : gof = ?

fog = ?

We have ;

f = { (1,2) , (3,4) , (2,2) }

Clearly ,

f(1) = 2

f(3) = 4

f(2) = 2

Also ,

g = { (2,1) , (3,1) , (4,2) }

Clearly ,

g(2) = 1

g(3) = 1

g(4) = 2

Now ,

We know that , (gof)(x) = g[f(x)]

Note :

Domain of (gof)(x) = Domain of f(x) .

Thus , for gof we will choose domain from f and we will get the range in g .

Thus ,

• (gof)(1) = g[f(1)] = g(2) = 1

• (gof)(2) = g[f(2)] = g(2) = 1

• (gof)(3) = g[f(3)] = g(4) = 2

Hence ,

gof = { (1,1) , (2,1) , (3,2) }

××××××××××××××××××××××××××××××××××××

Similarly ,

We know that , (fog)(x) = f[g(x)]

Note :

Domain of (fog)(x) = Domain of g(x) .

(x) .Thus , for fog we will choose domain from g and we will get the range in f .

Thus ,

• (fog)(2) = f[g(2)] = f(1) = 2

• (fog)(3) = f[g(3)] = f(1) = 2

• (fog)(4) = f[g(4)] = f(2) = 2

Hence ,

fog = { (2,2) , (3,2) , (4,2) }