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Asked: July 7, 20232023-07-07T23:47:48+05:30
2023-07-07T23:47:48+05:30In: Mathematics

Consider function f : A → B and g : B → C (A, B, C R) such that (gof)^–1 exists, then: (1) f and g both are one-one (2) f and g both are onto

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The correct option is

Af is one-one and g is ontoGiven that (gof)−1 exists. It means gof=g(f(x)) is one-one and onto.

Let h(x)=g(f(x)) and

h:A→C

Since h(x) is one-one, we have h(x1)=h(x2) for x1,x2∈A

⇒g(f(x1))=g(f(x2))⇒f(x1)=f(x2)

It means f must be one-one.

As h(x) is onto it means range of h(x) is C that is possible only when range of g(x) is also C.

Hence, g(x) is onto.