35-40. More composite functions Let f(x) = |x|, \(g(x)=x^{2}-4\), \(F(x)=\sqrt{x}\), and G(x) = 1/(x - 2). Determine the following composite functions and give their domains.
\(f\ \circ\ g\ \circ\ G\)
Step by step solution Step 1 of 1 Consider given functions f(x) = x , g(| |= x 4, F(x) = x and G(x) =1/(x 2) . The function f g G° °n be written as f (g(G(x))). First we must find the value of function g(G(x)). In order to do that we put the relation for G(x) in place of x in the function of g(x): g G = g(G(x)) ° g(x) = x 4 2 g(G(x)) = (G(x)) 4 g(G(x)) = ( 1 ) 4…………(1) x2 If we want to find the value of function f(x) for x =g(G(x)) we must use the function g(G(x)) in the relation for the function f(x): f ° ° = f(g(G(x))) f(x) = x | | f(g(G(x))) = g(G|x)) | Let’s use value for g(G(x)) from relation (1): | 2 | f(g(G(x))) = ( | x2) 4 | | | Hence the value of f g G is ( | 1 ) 4 .| ° ° |x2 |
