Question

Let $f:X\to Y$ and $g:Y\to X$ be two functions such that $\left(gof\right)\left(x\right)=x$ for all $xϵX$. Then

• A. $f$ is always onto
• B. $f$ is always one-one
• C. $g$ is always onto
• D. $g$ is always one-one

Answer 1

Answer: (B), (C)

$g$ is always onto
$f$ is always one-one

The given functions $f:X\to Y$ and $g:Y\to X$

Also $\left(gof\right)\left(x\right)=x$ Or $g\left(f\right(x\left)\right)=x$ for all $xϵR$

$g\left(f\right(x\left)\right)=x$, $f\left(x\right)=g^{-1}\left(x\right)$

For the function $f$ to be one-one, $f\left(a\right)=f\left(b\right)$, implies that $a=b$

Here, $g\left(f\right(x\left)\right)=x$,

$\to g\left(f\right(a\left)\right)=a$ ...$\left(1\right)$

$\to g\left(f\right(b\left)\right)=b$ ....$\left(2\right)$

From eq$\left(1\right)$ and $\left(2\right)$, we can see that if $f\left(a\right)=f\left(b\right),$ then $a=b$

From here the condition of one-one function is satisfying for $f\left(x\right)$ hence $f$ is always a one-one function.

We know that for function $f$ domain is $X$ and for function $g$ the domain is $Y$ $(\because f:X\to Y,g:Y\to X)$

As function $f$ is one-one, means for any value of $X$ there will only be one value of $Y$.

As $Y$ is the domain of function $g$,

Hence we can say that for every $Y$, there will be at least one $X$.

Thus the function $g$ is always an onto function.

So correct answers are $B$ and $C$.