Questions on Algebra: Quadratic Equation answered by real tutors!

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Question 155381: 1. True or false: The function "f(x) = 3^x" grows three times faster than the function "g(x) = x". Explain. : 1. True or false: The function "f(x) = 3^x" grows three times faster than the function "g(x) = x". Explain.
Answer by jim_thompson5910(9162) About Me  (Show Source):
You can put this solution on YOUR website!
Think about it this way.

If x=0, then f(0)=3^0=1 and g(0)=0.


If x=1, then f(1)=3^1=3 and g(1)=3.


If x=2, then f(2)=3^2=9 and g(2)=2.


If x=3, then f(3)=3^3=27 and g(3)=3.



So we have this table of values

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


From the table, we can see that g(x) increments by 1 as x increments by 1. On the other hand, we can see that f(x) 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 2/1=2. So from x=0 to x=1, f(x) is growing twice as fast as g(x).


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


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


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




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



So that means that the statement is false.