Question 1124192
f(x) = 3^x


the value of f(x) is whown in the following table.


<pre>

x       f(x)
0         1
1         3
2         9
3        27
4        82

</pre>


as can be seen, each time x goes up by 1, the value of f(x) goes up by a factor of 3.


take any random value of x and you'll see this tripling IN action.


i chose x = 7
that's a large number and will lead to equally large numbers.


f(x) = f(7) = 3^7 = 2187
f(x+1) = f(8) = 3^8 = 6561 = 3 * 2187 = 3^1 * 2187
f(x+2) = f(9) = 3^9 = 19683 3 * 6561 = 3 * 3 * 2187 = 3^2 * 2187
f(x+3) = f(10) = 3^10 = 59049 = 3 * 19683 = 3 * 3 * 3 * 2187 = 3^3 * 2187
f(x+4) = f(11) = 3^11 = 177147.


so, f(x + n) can be expected to be 3^n * 2187, if f(x) = 7


in general, then:


f(x + n) can be expected to be 3^n * f(x), for any value of x.


this works for any positive value of x.


take any random value that can be calculated, and you should see that it's true.


for example, i randomly chose x = 15.


f(15) = 3^15 = 14348907
f(15+5) = 3^20) = 3486784401
f(15+5) = 3^15 * 3^5 = 3486784401 = 3^20


but what happens when x is negative?


a simple example might show what happens.


let x = -3
then f(x) = x^-3 = 1/27


f(x+3) would be equal to f(-3+3) = f(0) which is equal to 1.


f(x+3) is also equal to f(-3) * 3^3 = 1/27 * 27 = 1


1 is equal to 27 * 1/27, therefore the formula holds.


another example:


f(-9) = 3^-9 = 5.080526343 * 10^-5


f(-9+5) = f(-9) * f(5) which is equal to 3^-9 * 3^5 which is equal to .012345679.


f(-9+5) is also equal to f(-4) which is equal to .012345679.


.012345679 divided by 5.080526343 * 10^-5 is equal to 243.


3^5 is equal to 243.


therefore, the relationship holds for positive and negative values of x.


consequently, f(x+n) is equal to f(x) * f(n).


in case it got lost in the discussion, however, the answer to your questions is:


a. for any value of x, f(x+1)is how many times as large as f(x). 


f(x+1) is 3^1 times as large as f(x) = 3 times as large.


b. for any value of x, f(x+2) is how many times as large as f(x). 


f(x+2) is 3^2 times as large as f(x) = 9 times as large.


c. If x increases by 4, f(x) becomes how many times as large?


f(x+4) is 3^4 times as large as f(x) = 81 times as large.