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# .)?

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Answer 1 · 2018-02-14T10:49:22

#4/3#

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#