Question 1124192: The function f(x)=3^x is an exponential function and is often referred to as the "tripling function."
a. for any value of x, f(x+1)is how many times as large as f(x).
b. for any value of x, f(x+2) is how many times as large as f(x).
c. If x increases by 4, f(x) becomes how many times as large?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! f(x) = 3^x
the value of f(x) is whown in the following table.
x f(x)
0 1
1 3
2 9
3 27
4 82
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.
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