Finding a Function of X to Satisfy Certain Conditions

[WhyIsItSo]

This should be a dead simple answer, but I don't know what this is called, or what to search on...

I need a function of x such that:

0 <= x <= 1

Where f(0) = 0, and f(1) = infinity.

I also need the curve to be relatively flat for "lower numbers", say up to 0.3, and begin a very sharp curve towards infinity around 0.8 or so.

A direct answer would be fine, but references would be just as welcome; I don't mind reading, just don't know where to point my browser at this point.

 

[arildno]

Well, I can make it exactly flat for you all the way up to 0.8:
For x less than 0.8, f(x)=0.
For x equal to or greater than 0.8, we let $$f(x)=A\frac{x-0.8}{1-x}$$, where A is a humungous number of your choice.

 

[uman]

f(x) = 0 if x=0
if 0<x<=1, f(x)=1/(1-x)

 

[uman]

f(x) = 0 if 0<=x<=0.3
f(x) = x if 0.3<x<=0.8
f(x) = (10000000000000000000000)!/(1-x) if 0.8<x

 

[WhyIsItSo]

uman: your equation looks like a simple way to get what I needed.

arildno: Your example gets me started on ways to shape uman's examples to suit my needs.

Thank you both.

 

[uman]

Out of curiosity, what do you need this for?

 

[WhyIsItSo]

Out of curiosity, what do you need this for?

I am building a MMOG. There are several applications for this equation, but the one in my mind at the time I made this request relates to "speed limits" for space vehicles in the game. While I am seeking a certain realism scientifically speaking, there are many reasons to depart from hard science; this is after all a science fiction product.

So, I wish to address energy requirements in some fashion for sub-light travel, as well as some formula for restricting Faster-Than-Light travel. Star Trek uses a Warp 10 limit as needing infinite energy. Successive Warp speeds are exponentially faster than preceding ones, with energy requirements rapidly climbing towards infinity as Warp 10 is approached.

I like the Star Trek approach, and want to base my "science" on that... but I needed a sharply rising exponent for my formula describing energy requirements from Warp 9 to Warp 10. I was not satisfied with that part in the Star Trek world.

I hope my reply doesn't disappoint you. It is a more or less trivial application

 

[CRGreathouse]

So, I wish to address energy requirements in some fashion for sub-light travel, as well as some formula for restricting Faster-Than-Light travel. Star Trek uses a Warp 10 limit as needing infinite energy. Successive Warp speeds are exponentially faster than preceding ones, with energy requirements rapidly climbing towards infinity as Warp 10 is approached.

I worked on precisely the same problem as a freshman in high school. I had on more data point to fit, from some episode that gave the relative speeds of two high 'warp' numbers. Unfortunately my idea of a good model at the time was a modified polynomial fit, which is a bad way to do it.

My recommendation is to set warp 1 to 1, warp 9 to w (which is whatever you need for your gameplay to work), warp x for 1 < x < 9 to $w^{(x-1)/8},$ and warp x for 9 < x < 10 to $w^{(10-x)^{-0.1}}.$ These are all in terms of the speed of light.