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I want to design a function that satisfies the following:

generally speaking, $f(x) = y$

$$f(5.51) = 1$$ $$f(95.31) = 200$$

How can I go about designing the function to satisfy these requirements?

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Hmm, what kind of shapes would you like it to be? Is it convex/concave? Polynomial? Any symmetries? Or would you like to fit the data into certain functions? [Always hit <kbd>ENTER</kbd>...] –  FrenzY DT. Aug 20 '12 at 13:40
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Next time please use TeX formatting as well as refraining from possibly offensive language. –  nbubis Aug 20 '12 at 13:41
    
Sure thing. I'd like this function to be exponential, or at least that general shape. –  Dan Aug 20 '12 at 13:42
    
The trouble is, the two "rules" f(5.51) = 1 and f(95.31) = 200 are obviously useless when x is not 5.51 or 95.31. Do you want us to guess what y is when x = 5.5? There are too little data points to use (non-linear) regression analysis‌​. –  William C Aug 20 '12 at 13:48
    
I seem to be confusing people about the purpose of my question. Basically, I'm designing this function so I can visually lay out elements in a web form such that they are aesthetically pleasing as well as precisely fitting numerical requirements. Whether or not that knowledge will please or enrage you is beyond me. –  Dan Aug 20 '12 at 14:01
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1 Answer 1

up vote 1 down vote accepted

If you want a function of the form $f(x)=a e^{bx}$ then simply solve for $a$ and $b$ using your data points.

If you want a polynomial function of degree $n$ then you'll need $n+1$ data points. With two data points, you get a polynomial of degree 1 (at most).

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Awesome. That's great. Unfortunately I can't vote this solution up because I don't have enough reputation. I'm also now wondering how could I do the same thing using a polynomial? –  Dan Aug 20 '12 at 13:48
    
@Dan, I've edited my answer. –  lhf Aug 20 '12 at 13:50
    
The last statement sounds good, y=mx+c will suffice here, (m,c are to be resolved from the given two conditions). –  lab bhattacharjee Aug 20 '12 at 13:51
    
Ah, cool. Let's add $$f(40) = 10$$ –  Dan Aug 20 '12 at 13:52
    
I mentioned "the given two conditions". With n given conditions, we need n degree polynomial, right? –  lab bhattacharjee Aug 20 '12 at 13:55
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