explicit
f(x) = 9*1.5^x
recursive
f(x) = {1.5*f(x-1) with f(0) = 9}
Let's write the recursive function first. We have 2 basic states. For f(0) we have the initial 9 grams of yeast. For f(n) where n > 0, we have 1.5*f(n-1). So
f(x) = {1.5*f(x-1) with f(0) = 9}
So if we solve it for f(2) we get
f(2) = 1.5*f(1)
f(1) = 1.5*f(0)
f(0) = 9
Since f(0) is 9, we can substitute that value into the equation for f(1), getting
f(1) = 1.5*f(0) = 1.5*9 = 13.5
And now we can plug that into the equation for f(2), giving
f(2) = 1.5*f(1) = 1.5*13.5 = 20.25
For the explicit function, we can rely of powers. 1.5^0 = 1; 1.5^1 = 1.5; 1.5^2 = 2.25; etc. So we get
f(x) = 9*1.5^x
Then if we evaluate f(2), we get
f(2) = 9*1.5^2
f(2) = 9*2.25
f(2) = 20.25
The advantage of the explicit function is that it doesn't require x to be an integer. And the yeast culture doesn't grown in a giant spurt every hour. So you can use the explicit function to tell you how much yeast you would have after 1.75 hours or for that matter any fractional amount of hours instead of just integer multiples of an hour.
