determine the inverse function of `f(x) = sqrt(x + 3)`

state the domain and range for its function & its inverse and sketch the graph of both the function and inverse

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Let g be the inverse function of f ,then by def. of inverse

`g(f(x))=f(g(x))=x`

Thus

`g(f(x))=x` (i)

let

`f(x)=sqrt(x+3)=y`

`g(f(x))=g(y)=x` (i)

because

`sqrt(x+3)=y`

`x+3=y^2`

`x=y^2-3` (ii)

from (i) and (ii)

`g(y)=y^2-3`

`Thus`

inverse of f(x) is g(x) where

`f(x)=sqrt(x+3)`

`g(x)=x^2-3`

domain of f=`{x :x>=-3 ,x inR}`

range of f=`{ x : x>=0 , x in R}`

domain of g =`{x : x>=0 ,x in R}`

range of g= { x , x greater or equal to -3 ,x is real no. }

Graph in red for the function f(x).

and green for the function g(x).

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