What are the sum and difference of the identity function and the reciprocal function ?

Explanation

Consider `f:R -> R:f(x)=x` and `g :R-{0} -> R:g(x)=frac{1}{x}`

dom `( f )` `∩` dom `( g ) = R ∩ R − { 0 } = R − { 0 }`

`∴(f+g):R-{0}→R:(f+g)(x)=f(x)+g(x)=(x+frac{1}{x})`.
`( f + g ) ( x ) = ( x + frac{1}{ x} )` for all `x ∈ R − { 0 }`.

And, `( f − g ) : R − { 0 } → R : ( f − g ) ( x ) = f ( x ) − g ( x ) = ( x − frac{1}{ x} )`

`(f-g)(x)=(x-frac{1}{x})` for all `x∈R-{0}`.