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AP EAMCET201823 Apr 2018Morning ShiftMathematicsFunctionsActual

If f: A B and g: B C are two functions such that g f: A C is a bijection, then which one of the following is always true?

Options

  1. Af and g are bijections
  2. Bf is an injection and g is a surjection
  3. Cf is a surjection and g is an injection
  4. Df is a bijection but g is not a bijection

Correct answer

B. f is an injection and g is a surjection

Step-by-step solution

If f: A B and g: B C are two function such that g f: A C is bijection. As, g f is one-one for x, y Let f(x)=f(y)g (f(x))=g (f(y)) g f(x)=g f(y) x=y f is one one. Now, Let Z C As gof is onto for every element Z C there exist y A that is in domain of gof. gathered g f(y)=Z g(f(y))=Z gathered f lies in range of B as f: A B . For a, y A, f(y) B , let f(y)=x . For every element Z C , we have at X B . Such that g(x)=Z . g is onto Hence, f is an injection and g is a surjection.

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