𝑔(. ) is a function from 𝐴 to 𝐵, 𝑓(. ) is a function from 𝐵 to 𝐶, and their composition defined as 𝑓(𝑔(. )) is a mapping from 𝐴 to 𝐶.
If 𝑓(. ) and 𝑓(𝑔(. )) are onto (surjective) functions, which ONE of the following is TRUE about the function 𝑔(. )?
𝑔(. ) must be an onto (surjective) function.
𝑔(. ) must be a one-to-one (injective) function.
𝑔(. ) must be a bijective function, that is, both one-to-one and onto.
𝑔(. ) is not required to be a one-to-one or onto function.
g: A→B
f: B → C are two functions
Hence fog: A→C is a composite function and also given that fog: A→C is an onto function
Hence f: B→C is necessary to be an onto function, where as g: A→B need not be onto (or) one-one.
Let as take example
Here gof: A→C an onto, but g: A→B is neither one-one nor onto.