SOLUTION: I need to find the functions (f o g), (g o f),(f o f), and (g o g) and their domains for: 34. f(x) = x^2, g(x) = sqrt(x-3) 38. f(x) = 1/sqrt(x), g(x) = x^2

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Question 33362This question is from textbook College Algebra
: I need to find the functions (f o g), (g o f),(f o f), and (g o g) and their domains for:
34. f(x) = x^2, g(x) = sqrt(x-3)
38. f(x) = 1/sqrt(x), g(x) = x^2 - 4x
Thank you very much! I know I'm wrong becasue for 34 (g o f) I came out with an imaginary number.
This question is from textbook College Algebra

Answer by mukhopadhyay(490) (Show Source):
You can put this solution on YOUR website!
f(x)=x^2, g(x)=sqrt(x-3) ;
(f o g) = f(g(x)) = f(sqrt(x-3)) = (x-3); Domain: x >= 3;
(g o f) = g(f(x)) = g(x^2) = (x^2-3); Domain: x is a real number;
(f o f) = f(f(x)) = f(x^2) = x^4; Domain: x is a real number;
(g o g) = g(g(x)) = g(sqrt(x-3)) = Sqrt[sqrt(x-3)-3]; Domain: x>= 3
.........
f(x)=1/sqrt(x); g(x)=(x^2-4x)=x(x-4);
(f o g) = f(g(x)) = f(x^2-4x) = 1/[sqrt(x)(x-4)]; Domain: x < 0 or x>= 4;
(g o f) = g(f(x)) = g(1/sqrt(x)) = 1/x - 4/sqrt(x); Domain: x > 0 ;
(f o f) = f(f(x)) = f(1/sqrt(x)) = 1/[sqrt(1/sqrt(x))] = x^(1/4); Domain: x>0 ;
(g o g) = g(g(x)) = g(x^2-4x) = (x^2-4x)^2 - 4(x^2-4x) = (x^2-4x)(x^2-4x-4)
= x(x-4)(x^2-4x-4); Domain: x is a real number

SOLUTION: I need to find the functions (f o g), (g o f),(f o f), and (g o g) and their domains for: 34. f(x) = x^2, g(x) = sqrt(x-3) 38. f(x) = 1/sqrt(x), g(x) = x^2 - 4x Thank you