Question 3.5.9: Find the sum and the difference of the identity function and......

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

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

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

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