Question

I: If $f:A\to B$ is a bijection only then does $f$ have an inverse function
II: The inverse function $f:R^{+}\to R^{+}$ defined by $f\left(x\right)=x^2$ is $f^{-1}\left(x\right)=\sqrt{x}$

• A. only I is true
• B. only II is true
• C. both I and II are true
• D. neither I nor II true

Answer 1

The correct option is C both I and II are true

$f$ is a bijection $\Rightarrow$ $f$ is one one and onto.
$\therefore$ there exists on inverse value for every value in co-domain.
$\therefore$ $f$ is invertible if and only if $f$ is a bijection
$f:R^{+}\to R^{+};f\left(x\right)=x^2=y$
$\Rightarrow x=\sqrt{y}=f^{-1}\left(y\right)$