Question 3.5.13: Find the quotient of the identity function by the reciprocal......

Let f: R \rightarrow R: f(x) and g: R-\{0\} \rightarrow R: g(x)=\frac{1}{x} be the identity function and the reciprocal function respectively.

Now, \operatorname{dom}\left(\frac{f}{g}\right)=\operatorname{dom}(f) \cap \operatorname{dom}(g)-\{x: g(x)=0\}

and \{x: g(x)=0\}=\left\{x: \frac{1}{x}=0\right\}=\phi .

\therefore \quad \operatorname{dom}\left(\frac{f}{g}\right)=[R \cap R-\{0\}]-\phi=R-\{0\} .

So, \frac{f}{g}: R-\{0\} \rightarrow R:\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}=\frac{x}{\frac{1}{x}}=x^{2} .

Hence, \left(\frac{f}{g}\right)(x)=x^{2} for all x \in R-\{0\} .