SOLUTION: Listed below are 5 functions each denoted g(x) and each involving a real number x constant c>1. If f(x)=2 , which of these 5 functions yeilds the greates

Algebra ->  Proportions -> SOLUTION: Listed below are 5 functions each denoted g(x) and each involving a real number x constant c>1. If f(x)=2 , which of these 5 functions yeilds the greates      Log On


   

Question 201539: Listed below are 5 functions each denoted g(x) and each involving a real number
x
constant c>1. If f(x)=2 , which of these 5 functions yeilds 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 x
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29+=+2%5Ex means this function will take whatever you give it and raise 2 to that power. So when we give it g(x) it will raise 2 to that power: f%28g%28x%29%29+=+2%5E%28%28g%28x%29%29%29

So now we can see what will happen to this for the various definitions of g(x).
A. g(x)= cx
f%28cx%29+=+2%5E%28cx%29
B. g(x)= c/x
f%28c%2Fx%29+=+2%5E%28c%2Fx%29+=+2%5E%28c%2A%281%2Fx%29%29
C. g(x)= x/c
f%28x%2Fc%29+=+2%5E%28x%2Fc%29+=+2%5E%28x%2A%281%2Fc%29%29
D. g(x)= x-c
f%28x-c%29+=+2%5E%28%28x-c%29%29
E. g(x)= log x
f%28log%28%28x%29%29%29+=+2%5E%28log%28%28x%29%29%29

Now we just need to figure out which exponent is largest
When c>1 and x>1 ...
A. cx > x and cx > c
B. 1/x is between 0 and 1 so c*(1/x) < c.
C. 1/c is between 0 and 1 so x*(1/c) < x.
D. x-c < x
E. log(x) < x. This is the trickiest one to explain. If you look at graphs of y=x and y=log(x) superimposed on each other you will see that for all x-values the graph of graph of y=log(x) is below the graph of y=x. So log(x) < x for all x. (Unfortunately I can't get Algebra.com software to graph log(x). If I could I'd show you these graphs here.)

In summary, when c>1 and x>1 only c*x is guaranteed to be larger than both c and x. So c*x is the largest exponent which means 2%5E%28cx%29 is the largest which means the answer is A.