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If f(x) = 2x, which of these 5 functions yields the greatest value for f(g(x)), for all x > 1

Listed below are 5 functions, each denoted g(x) and each involving a real number constant
c > 1.If f(x) = 2x, which of these 5 functions yields the greatest value for f(g(x)), for all
x > 1?
 
 
A. g(x) = cx
B. g(x) = c/x
C. g(x) = x/c
D. g(x) = x – c
E. g(x) = logcx
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1 Answer

f(x)=2x
 
Then f(g(x))= 2g(x)
 
we are supposed evalute the 5 expressions to determine which would yield the highest value of f(x).  Because the g(x) functions share a common base, we can evalute which one would have the highest value, and that would yeild the highest value for f(x).
 
For any value of c,  answer A. is clearly larger than a fraction (B. and C.), and larger than D.  The only sticky point is answer E, is cx larger than logcx?
 
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In general the expression:  logab=n can be written as an=b 
For example:  let a = 2 and b=8.  We know that n=3 because 8 is 23.  Similarly, log28=3.  
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returning to the problem, we know that the value of logcx must be smaller than x for any c, therefore it is less than "cx" as well.
  So the answer is:  A.  g(x)=cx, yeilds the highest value of f(g(x)) for all x>1.
 
 

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