Question

Suppose f is a function of the form
f(x) = a^{bx}
for some constants a and b. The doubling time for f is 1, i.e.,
f(x+1) = 2f(x)
for all x.
Then f(x) =

Answer 1

Just plug and chug your formula f(x)=a^(bx) into with your doubling time formula. On the left you're substituting x+1 for x and on the right you're multiplying it by 2. Take a guess and we'll help you get there!

Answer 2

Interestingly, you start out with 2 variables but you might find that you get an answer in terms of two variables that allows you to directly plug them in. So you don't actually need to explicitly find a or b at all!

Answer 3

f(x) = a^(bx)
Try a few values of x such as x = 0, x = 1, x = 2, etc.
f(0) = a^(b*0) = a^(0) = 1
f(1) = a^(b*1) = a^b = 2 (because it doubles the previous value)
f(2) = a^(b*2) = a^(2b) = 4
f(3) = a^(b*3) = a^(3b) = 8

You can see the pattern:
f(0) = 2^0
f(1) = 2^1
f(2) = 2^2
f(3) = 2^3

Therefore, a = 2 and b = 1
f(x) = 2^x

Answer 4

so there is now specific way to find the values, just have to guess?

Answer 5

f(x+1)=2f(x) right? so we have f(x)=a^(bx) so...
\[a^{b(x+1)}=2a^{bx}\]use exponent rules...\[a^{bx}*a^b=2a^{bx}\]
divide both sides by a^(bx) and you get:
\[a^b=2\]

Now back to the original f(x)=a^(bx) well that's easily rewritten as: \[f(x)=a^{bx}=(a^b)^x=2^x\]

So there you go, f(x)=2^x

Answer 6

If you need help following the steps I skipped, I can write them all out for you. =D

Answer 7

thats okay i got it now. Thank you

Answer 8

Coolio. B)

Answer 9

how would you do the math if it was x+3 instead of x+1?

Answer 10

Almost identically, but plug in a 3 instead of a 1. The term on the left would be a^{b(x+3)} so you'd get a^(bx)*a^(3b) and so you'd end up with (a^b)^3=2 so you would end up plugging in a^b=cuberoot(2) for a^b to get f(x)=cuberoot(2)^x or 2^(x/3)=f(x)

Answer 11

okay thank you very much!