Question 3.5.11:  Find the product of the identity function by the reciprocal......

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

Then, \operatorname{dom}(f g)=\operatorname{dom}(f) \cap \operatorname{dom}(g)=R \cap R-\{0\}=R-\{0\} .

\therefore \quad(f g): R-\{0\} \rightarrow R:(f g)(x)=f(x) \cdot g(x)=\left(x \times \frac{1}{x}\right)=1 .

Hence, (f g)(x)=1 for all x \in R-\{0\} .