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

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Answer 1 · 2017-07-20T03:32:09

Please see below.

The domain of a composition #f @ g# is the part of domain(#g#) for which #g(x)# is in domain(#f#).

In this question #g(x) = sqrtx# has domain #[0,oo)#. so the domain of the composition can't be any bigger than that.

#f(x) = x^2# has domain all real numbers. So every #g(x)# is in the domain of #f#

The domain of #f@g# is #[0,oo)#.

The rule

#(f@g)(x) = f(g(x))#

Replace #g(x)# by #sqrtx#

#(f@g)(x) = f(sqrt(x))#

For any input #(u)#, #f# gives #(u)^2#, so

#f(sqrtx) = (sqrtx)^2#

But #(sqrtx)^2 = x#

So we finish with

#(f@g)(x) = x# where #x# is in #[0.oo)#