Question:
Let f: R → R be defined as f(x)
(A) $f$ is one-one onto (B) $f$ is many-one onto
(C) $f$ is one-one but not onto (D) $f$ is neither one-one nor onto
Solution:
$f: \mathbf{R} \rightarrow \mathbf{R}$ is defined as $f(x)=3 x$.
Let $x, y \in \mathbf{R}$ such that $f(x)=f(y)$.
$\Rightarrow 3 x=3 y$
$\Rightarrow x=y$
∴f is one-one.
Also, for any real number $(y)$ in co-domain $\mathbf{R}$, there exists $\frac{y}{3}$ in $\mathbf{R}$ such that $f\left(\frac{y}{3}\right)=3\left(\frac{y}{3}\right)=y$.
∴f is onto.
Hence, function f is one-one and onto.
The correct answer is A.