Let f and g be real functions, defined by `f(x)=sqrt(x-1)andg(x)=sqrt(x+1)`. Find (i) `(f+g)(x)` (ii) `(f-g)(x)` (iii) `(fg)(x) `(iv) `((f)/(g))(x)`.

Let f and g be real functions, defined by `f(x)=sqrt(x-1)andg(x)=sqrt(

1.	Let f and g be real functions, defined by `f(x)=sqrt(x-1)andg(x)=sqrt(x+1)`. Find (i) `(f+g)(x)` (ii) `(f-g)(x)` (iii) `(fg)(x) `(iv) `((f)/(g))(x)`.
Answer» Clearly, `f(x)sqrt(x-1)` is defined for all real values of x for which `x-1ge0,i.e.,xge1."So, dom "(f)=[1,oo)`. Also, `g(x)=sqrt(x+1)` is defined for all real values of x for which `x+1ge0,i.e.,xge-1."So, dom "(g)=[-1,oo)`. `:."dom "(f)nn"dom "(g)={1,oo)nn[-1,oo)=[1,oo)`. (i) `(f+g):[1,oo)toR` is given by `(f+g)(x)=f(x)+g(x)=(sqrt(x-1)+sqrt(x+1))`. (ii) `(f-g):[1,oo)toR` is given by `(f-g)(x)=f(x)-g(x)=(sqrt(x-1)-sqrt(x+1))` (iii) `(fg):[1,oo)toR` is given by `(fg)(x)=f(x).g(x)=sqrt(x-1)xxsqrt(x+1)=sqrt(x^(2)-1)`. (iv) `{x:g(x)=0}={x:sqrt(x+1)=0}={x:x+1=0}={-1}`. `:."dom "(f)nn"dom "(g)-{x:g(x)=0}` `=[1,oo)nn[-1,oo)-{-1}=[1,oo)`. `:.(f)/(g)to[1,oo)toR` is given by `((f)/(g))(x)=(f(x))/(g(x))=sqrt(x-1)/sqrt(x+1)`.