Question

Let f : R → R be defined by $f\left(x\right)=\frac{1}{x}.$ Then, f is
(a) one-one
(b) onto
(e) bijective
(d) not defined

Answer 1

Given: The function f : R → R be defined by $f\left(x\right)=\frac{1}{x}.$

To check f is one-one:

$$\begin{gathered} \mathrm{Let}f\left(x_1\right)=f\left(x_2\right) \\ \Rightarrow \frac{1}{x_1}=\frac{1}{x_2} \\ \Rightarrow x_1=x_2 \\ \mathrm{Hence},f\mathrm{is}\mathrm{one}-\mathrm{one}. \\ \mathrm{To}\mathrm{check}\mathit{}f\mathrm{is}\mathrm{onto}: \\ \mathrm{Since},y=\frac{1}{x} \\ \Rightarrow x=\frac{1}{y} \\ \Rightarrow y\in R-\left\{0\right\}\ne R \\ \mathrm{There}\mathrm{is}\mathrm{no}\mathrm{pre}-\mathrm{image}\mathrm{of}y=0. \\ \mathrm{Hence},f\mathrm{is}\mathrm{not}\mathrm{onto}. \end{gathered}$$

Hence, the correct option is (a).

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