Let f and g be real functions, defined by `f(x)=sqrt(x+2)andg(x)=sqrt(4-x^(2))`. Find (i) (f+g)(x) (ii) (f-g)(x) (iii)(fg)(x) (iv) `(ff)(x)` (v) (gg)(x) (vi) `((f)/(g))(x)`.

Let f and g be real functions, defined by `f(x)=sqrt(x+2)andg(x)=sqrt(

1.	Let f and g be real functions, defined by `f(x)=sqrt(x+2)andg(x)=sqrt(4-x^(2))`. Find (i) (f+g)(x) (ii) (f-g)(x) (iii)(fg)(x) (iv) `(ff)(x)` (v) (gg)(x) (vi) `((f)/(g))(x)`.
Answer» Clearly, `f(x)=sqrt(x+2)` is defined for all `x""inR` such that `x+2ge0,i.e.,xge-2`. `:."dom "(f)=[-2,oo)`. Again, `g(x)=sqrt(4-x^(2))` is defined for all `x""inR` such that `4-x^(2)ge0`. But, `4-x^(2)ge0impliesx^(2)-4le0implies(x+2)(x-2)leimpliesx""in[-2,2]`. `:."dom "(g)=[-2,2]` `:."dom "(f)nn"dom "(g)=[-2,oo)nn[-2,2]=[-2,2]`. (i) `(f+g):[-2,2]toR` is given by `(f+g)(x)=f(x)+g(x)=sqrt(x+2)-sqrt(4-x^(2))` (ii) `(f-g):[-2,2]toR` is given by `(f-g)(x)=f(x)=g(x)=sqrt(x+2)-sqrt(4-x^(2))` ltbgt (iii) `(fg):[-2,2]toR` is given by `(fg)(x)=f(x).g(x)=(sqrt(x+2))(sqrt(2-x^(2)))` `=sqrt((x+2)^(2)(2-x))=(x+2)sqrt((2-x))`. (iv) `(ff):[-2,2]toR` is given by `(ff)(x)=f(x)_.f(x)=(sqrt(x+2))(sqrt(x+2))=(x+2)`. (v) `(gg):[-2,2]toR` is given by `(gg)(x)=g(x).g(x)=(sqrt(4-x^(2)))(sqrt(4-x^(2)))=(4-x^(2))`. (vi) `{x:g(x)=0}={x:4-x^(2)=0]={x:(2-x)(2+x)=0}={-2,2}` `"dom "((f)/(g))="dom "(f)nn"dom "{g}-{x:g(x)=0}` `=[-2,2]-[-2,2]=(-2,2)`. `:.(f)/(g):(-2,2)toR` is given by `((f)/(g))(x)=(f(x))/(g(x))=(sqrt(x+2))/(sqrt(4-x^(2)))=(sqrt(2+x))/((sqrt(2+x))(sqrt(2-x)))=(1)/((sqrt(2-x)))`.