Question

Question from Function!

Are f and g both necessarily onto, if gof is onto?

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Answer 1

No its not necessary for g and f to be both onto for gof is onto as onto cant be determined like this, for onto a specific range is provided and if the gof lies between the provided range it can be onto as if

F=sinx and G= |x|

And it is given that R- [0, 1]

Then gof comes |sinx|

and it is onto as it lies between [0,1]

But g and f are not both onto.

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Answer 2

Explanation:

let

f : A --> B

g : B --> C

let

C belongs to c

since gof : A --> C , A belongs to a

gof(a) = c => g(f(a)) = c

g : B --> C ,

B belongs to f(a) for any C belongs to c

there exist an element belonging to B

hence g is onto but f is not onto .