# 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
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If , then and .

If , then and .

If , then and .

If , then and .

So we have this table of values

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

From the table, we can see that increments by 1 as x increments by 1. On the other hand, we can see that 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.

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

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

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

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

Note: the function does however grow three times faster than , but that is for another problem.

So that means that the statement is false.