SOLUTION: True or false: The function "f(x) = 3^x" grows three times faster than the function "g(x) = x". Explain

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Question 172663: True or false: The function "f(x) = 3^x" grows three times faster than the function "g(x) = x". Explain
Answer by jim_thompson5910(28715) About Me  (Show Source):
You can put this solution on YOUR website!
Think about it this way.

If x=0, then f%280%29=3%5E0=1 and g%280%29=0.


If x=1, then f%281%29=3%5E1=3 and g%281%29=3.


If x=2, then f%282%29=3%5E2=9 and g%282%29=2.


If x=3, then f%283%29=3%5E3=27 and g%283%29=3.



So we have this table of values

xf(x)g(x)
010
131
292
3273


From the table, we can see that g%28x%29 increments by 1 as x increments by 1. On the other hand, we can see that f%28x%29 goes from 1 to 3 (a difference of 2), 3 to 9 (a difference of 6), 9 to 27 (a difference of 18), etc. So the differences between each term is: 2, 6, 18, etc....


This means that from x=0 to x=1, the average rate of change for g(x) is 2. From x=1 to x=2, the average rate of change for g(x) is 6. From x=2 to x=3, the average rate of change for f(x) is 18.

-------------------

So dividing the first average rate of change 2 by 1, we get 2%2F1=2. So from x=0 to x=1, f%28x%29 is growing twice as fast as g%28x%29.


Dividing the second average rate of change 6 by 1, we get 6%2F1=6. So from x=1 to x=2, f%28x%29 is growing six times as fast as g%28x%29.


Dividing the third average rate of change 18 by 1, we get 18%2F1=18. So from x=2 to x=3, f%28x%29 is growing eighteen times as fast as g%28x%29.


As you can see, the exponential function is not growing at a constant rate. So f%28x%29 cannot be growing 3 times faster than g%28x%29




Note: the function f%28x%29=3x does however grow three times faster than g%28x%29=x, but that is for another problem.



So that means that the statement is false.