Question

If $f:A\to B,g:B\to A$ be two functions such that $gof=I_A$, then which among following is/are correct?

• A. $f$ is many-one function.
• B. $g$ is an into function.
• C. $g$ is a surjective function.
• D. $f$ is an injective function.

Answer 1

Answer: (C), (D)

$f$ is an injective function.

Let $x,y\in A$ such that $f\left(x\right)=f\left(y\right),$ then
$\begin{array}{c}f\left(x\right)=f\left(y\right)\\ \Rightarrow g\left(f\right(x\left)\right)=g\left(f\right(y\left)\right)\\ \Rightarrow gof\left(x\right)=gof\left(y\right)\\ \Rightarrow I_A\left(x\right)=I_A\left(y\right)\\ \Rightarrow x=y\end{array}$
$\therefore f$ is One-one function
Let $x$ be an arbitrary element of $A$.
As $f:A\to B$ is a function,
$\therefore f\left(x\right)\in B$
Let $f\left(x\right)=y$
$g\left(y\right)=g\left(f\right(x\left)\right)$
$=gof\left(x\right)$
$=I_A\left(x\right)$
$=x$
$\therefore$ For every $x\in A,y=f\left(x\right)\in B$ such that $g\left(y\right)=x$
Hence $g$ is onto function.