Suppose that f ( x ) and g ( x ) are functions which satisfy f ( g ( x ) ) = x^2 and g ( f ( x ) ) = x^3 for all x ≥ 1 . If g ( 16 ) = 16 , then compute log_2 g ( 4 ) . (You may assume that f ( x ) ≥ 1 and g ( x ) ≥ 1 for all x ≥ 1 .)?

1 Answer
Feb 14, 2018

4/3

Explanation:

Consider g(f(g(4)))

Since f(g(4))=4^2=16, this is equal to g(16)=16.

On the other hand, since g(f(x))=x^3, we see that this is (g(4))^3.

So

(g(4))^3=16 implies 3 log_2(g(4)) = log_2 16 = 4