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 $x \in R-\{0\}$.