SOLUTION: given the functions f(x)=x+2 and g(x)=x^2, determine all values of x for which f(g(x))=g(f(x)).

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Question 1194123: given the functions f(x)=x+2 and g(x)=x^2, determine all values of x for which f(g(x))=g(f(x)).
Answer by ikleyn(52777) About Me  (Show Source):
You can put this solution on YOUR website!
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given the functions f(x)=x+2 and g(x)=x^2, determine all values of x for which f(g(x))=g(f(x)).
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Left side of this equation is

    f(g(x)) = g(x) + 2 = x^2 + 2.


Right side of this equation is

    g(f(x)) = (f(x))^2 = (x+2)^2 = x^2 + 4x + 4.


So, this equation  f(g(x))=g(f(x))  takes the form

    x^2 + 2 = x^2 + 4x + 4.


Cancel the terms x^2 in both sides; keep the term with x in right side; collect constant terms in left side

    2 - 4 = 4x

      -2  = 4x

      -1/2  =  x.


ANSWER.  x = -1/2.

Solved.