Question 3.5.12: Find the quotient of the identity function by the modulus fu......

Let f: R \rightarrow R: f(x)=x and g: R \rightarrow R: g(x)=|x| be the identity function and the modulus 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\}=\{x:|x|=0\}=\{0\} .

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

So, \frac{f}{g}: R-\{0\} \rightarrow R:\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}=\frac{x}{|x|}=\left\{\begin{array}{l}1, \text { when } x>0 \\ -1, \text { when } x<0.\end{array}\right.

Hence, \left(\frac{f}{g}\right)(x)=\left\{\begin{array}{l}1, \text { when } x>0 \\ -1, \text { when } x<0 .\end{array}\right.