Question

If the function $g o f$ is defined and onto, then

• A. neither $f$ nor $g$ is onto
• B. $f$ and $g$ both are necessarily onto
• C. $f$ must be onto
• D. None of the above

Answer 1

Answer: (D)

Consider $ f:\{1,2,3,4\} \rightarrow\{1,2,3,4\} $ and $g:\{1,2,3,4\} \rightarrow\{1,2,3\} $ defined as $f(1)=1, f(2)=2$, $f(3)=f(4)=3, g(1)=1, g(2)=2$ and $g(3)=g(4)=3$. It can be seen that gof is onto but $f$ is not onto.