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

Question

Listed below are 5 functions, each denoted g(x) and each involving a real number constant
c > 1.If f(x) = 2^x, 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) = log_cx

Answer 1

f(x)=2^x

 

Then f(g(x))= 2^g(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 log_cx?

 

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In general the expression:  log_ab=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 log_cx 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.