​ 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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Asked: 2023-07-07T23:47:48+05:30 In: 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

Hi, I have a question and I hope anyone could answer it:

Consider function f : A → B and g : B → C (A, B, C (subseteq) R) such that (gof)–1 exists, then:

(1) f and g both are one-one

(2) f and g both are onto

(3) f is one-one and g is onto

(4) f is onto and g is one-one

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    1 Answer

    2. The correct option is A f is one-one and g is onto
      Given 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.

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