SOLUTION: Hi I'm confused with his question. However, my answer is below. True or false: The function f(x)=3^x grows three times faster than the function g(x)=x. Explain. MY answer is

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Hi I'm confused with his question. However, my answer is below. True or false: The function f(x)=3^x grows three times faster than the function g(x)=x. Explain. MY answer is       Log On


   

Question 150404: Hi
I'm confused with his question. However, my answer is below.
True or false: The function f(x)=3^x grows three times faster than the function g(x)=x. Explain.
MY answer is true because F(x)=3^x is exponential and grows faster because what ever the x (exponent) is, it multiplies that 3, that many times over. And the g(x)=x is one number muliplied with another number. Is this correct or am I thinking of it the wrong way? Please help, I'm the only one who has this different answer and I don't understand why they all have false.
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
True or false: The function f(x)=3^x grows three times faster than the function g(x)=x. Explain.
MY answer is true because F(x)=3^x is exponential and grows faster because what ever the x (exponent) is, it multiplies that 3, that many times over. And the
g(x)=x is one number multiplied with another number. Is this correct or am I thinking of it the wrong way? Please help, I'm the only one who has this different answer and I don't understand why they all have false.
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Comparing g(x) = x to f(x)=3^x
g(x) = x is a line with slope=1 and y-intercept = 0
As x increases, y increases
It looks like this:
graph%28400%2C300%2C-10%2C10%2C-10%2C10%2Cx%29
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y= 3^x is exponential with y-intercept at (0,1) and a constantly
increasing slope. It looks like this:
graph%28400%2C300%2C-10%2C10%2C-10%2C20%2C3%5Ex%29
--------
Note: It is incorrect to say "the exponent multiplies the three".
-----------------
Cheers,
Stan H.

Answer by jim_thompson5910(35256) 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.