For 52 only show me fgx which is also written f o g and gfx

(For #52 only show me (f(g(x)) which is also written (f o g) and g(f(x)) which is also written (g o f) . Clearly show how you got the domain for each)

Solution

( f o g)(x)

f(x) = x^2 , g(x) = sqrt( x-3)

To find ( f o g)(x) ; plug x= g(x)= sqrt(x -3) in f(x)

( f o g)(x) = ( sqrt(x-3) )^2 = x - 3

Domain : ( - inf , inf)

( g o f )(x) = g( f(x)) ; Plug x= x^2 in g(x)

= sqrt( x^2 - 3)

Domain x^2 -3 should be grtear than equal to zero

x^2 - 3 >=0

(x+ sqrt3)(x - sqrt3) .>=0

x>= sqrt3 ; x<= -sqrt3

Domain : ( - inf ,sqrt3] U [ sqrt3 , inf)