Seeing as this is posted under function composition, I'll treat
#(f * g)(x)# as the composition #f(g(x))#.
The domain of #f# is #RR#, since any real #x# can be squared.
The domain of #g# is #[0, +infty)# because roots only accept non negative input.
The composition is defined, when #x# is in the domain of #g#, and #g(x)# is in the domain of #f#. In other words:
With #f(g(x))#, #x# is the input (independent variable) for #g#, and #g(x)# is the input for #f#. Therefore, the inputs each must be part of the domains of the functions they are to be used in. We have the following constraints:
#x# must be non-negative, since it goes into the function #g(x) = sqrtx#.
#g(x)# must be real (it obviously is).
Therefore,
#f(g(x)) = (sqrtx)^2 = x#, for #x >= 0#.
