Let #f(x)=x^2# and #g(x)=sqrtx#, how do you find the domain and rules of #(f*g)(x)#?

Question

Lincoln Anderson · Accepted Answer

#f(g(x)) = x, x>=0#.

Given that this is under the function composition section, I'll handle #(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). Consequently, #f(g(x)) = (sqrtx)^2 = x#, for #x >= 0#.