Describe fog and gof wherever is possible for the following functions (i) `f(x)=sqrt(x+3),g(x)=1+x^(2)` (ii) `f(x)=sqrt(x),g(x)=x^(2)-1`

Describe fog and gof wherever is possible for the following functions

1.	Describe fog and gof wherever is possible for the following functions (i) `f(x)=sqrt(x+3),g(x)=1+x^(2)` (ii) `f(x)=sqrt(x),g(x)=x^(2)-1`
Answer» (i) Domain of `f` is `[3,oo)`, range of `f` is `[0,oo)` Domain is `g` is `R`, range of `g` is `[1,oo)` For go`f(x)` Since range of `f` is a subset of domain of `g` `:.` domain of gof is `[-3,oo)` {equal to the domain of `f`} `gof(x)=g{f(x)}=gsqrt(x+e)=1+(x+3)=x+r`. Range of gof is `[1,oo)` For `fog(x)` Since range of `g` is a subset fo domain of `f` `:.` domain of fog is `R` {equal to the domain of `g`} `fog(x)=f{g(x)}=f(1+x^(2))=sqrt(x^(2)+r` Range of fog is `[2,oo)` (ii) `f(x)=sqrt(x),g(x)=x^(2)-1` Domain of `f` is `[0,oo)` range of `f` is `[0,oo)` Domain of `g` is `R` range of `g` is`[-1,oo)` For `gof(x)` Since range of `f` is a subset of the domain of `g` `:.` domain of gof is `[0,oo)` and `g{f(x)}=g(sqrt(x))=x-1`. Range of gof is `[-1,oo)` For `fog(x)` Since range of `g` is not a subset of the domain of `f` i.e. `[-1,oo)cancelsub[0,oo)` `:.` fog is not defined on whole of the domain of `g` Domain of fog is `{xepsilonR` the domain of `g:g(x)epsilon[0,oo)`, the domain of `f}`. Thus the domain of fog is `D={xepsilonR:0leg(x)ltoo}` i.e. `D={xepsilonR:lex^(2)-1}={xepsilonR:xle-1` or `xge1}=(-oo,-1]uu[1,oo)` `fog(x)=f{g(x)}=f(x^(2)-1)=srt(x^(2)-1)` Its range is `[0,oo)`