Question:
Let $f: R \rightarrow R$ be defined by $f(x)=\frac{1}{x}$. Then, $f$ is
(a) one-one
(b) onto
(e) bijective
(d) not defined
Solution:
Given: The function $f: R \rightarrow R$ be defined by $f(x)=\frac{1}{x}$.
To check f is one-one:
Let $f\left(x_{1}\right)=f\left(x_{2}\right)$
$\Rightarrow \frac{1}{x_{1}}=\frac{1}{x_{2}}$
$\Rightarrow x_{1}=x_{2}$
Hence, $f$ is one-one.
To check $f$ is onto:\
Since, $y=\frac{1}{x}$
$\Rightarrow x=\frac{1}{y}$
$\Rightarrow y \in R-\{0\} \neq R$
There is no pre-image of $y=0$.
Hence, $f$ is not onto.
Hence, the correct option is (a).