Question

Let $f:A\to B$ and $g:B\to C$ be two functions and $g\circ f:A\to C$ is defined. Then which of the following statements is true?

• A. If $g\circ f$ is onto, then $f$ must be onto.
• B. If $f$ is into and $g$ is into, then $g\circ f$ must be onto function.
• C. If $g\circ f$ is one-one, then $g$ is not necessarily one-one.
• D. If $f$ is injective and $g$ is surjective, then $g\circ f$ must be bijective mapping.

Answer 1

The correct option is C If $g\circ f$ is one-one, then $g$ is not necessarily one-one.


Here, $g\circ f$ is onto but $f$ is not onto.


Here, $f$ is into and $g$ is into but $g\circ f$ is not onto function.


Here, $g\circ f$ is one-one and $g$ is not one-one.


Here, $f$ is injective and $g$ is surjective but $g\circ f$ is not a bijective mapping.

Hence, if $g\circ f$ is one-one, then $g$ is not necessarily one-one.