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I am working on an algorithm for my work and we are using an exponential function where $x =$ the users age and $f(x) = a$ value that will be subtracted from the users compatibility index.

I havent worked with calculus in quite a while so finding an answer to this problem has been imposible. Basically I need an equation that fits the following rules.

  1. vertical asymptote at $x = 100$ ($100$ is the maximum age that can be put in)
  2. graph is a positive function (in the first quadrant)
  3. At $x=55$ ($y$ unknown), a tangent line touches the curve; the slope of this tangent line must equal 1.
  4. $f(18) = 0$

EDIT: I have another problem as well. SImilar situation.

  1. vertical asymptote at $x = 18$ ($18$ is the maximum age that can be put in)
  2. graph is a negative function (in the first quadrant)
  3. At $x=8$ ($y$ unknown), a tangent line touches the curve; the slope of this tangent line must equal -1.
  4. $f(0) = 100$
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You have problem: exponential functions do not have vertical asymptotes. Even a sum of exponential functions of the form $\sum a_ie^{b_ix}$ is finite for all real $x$. –  Fly by Night Jan 23 '13 at 19:59
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1 Answer

up vote 1 down vote accepted

If you aren't worried about the function being defined when $x>100$, then you can start with the function $$g_1(x)=-\frac1x$$ to get a workable right general shape. Next, shift it to the right by $100$ to get the correct vertical asymptote--yielding $$g_2(x)=-\frac1{x-100}.$$ This satisfies the first two conditions. To satisfy the third condition, scale by a factor of $A$ to get $$g_3(x)=-\frac{A}{x-100}$$ (You'll need to set $g_3'(55)=1$ and solve to find the appropriate $A$.) Finally, shift vertically by some constant $C$ to get $$g_4(x)=-\frac{A}{x-100}+C$$ (You'll need to set $g_4(18)=0$ to solve for $C$.) Restrict $g_4$ to the interval $[0,100)$ to get the desired function $f$.

If you require that the function be defined for $x>100$, then this won't work. Let me know, if so.

Edit: Apparently, you are looking for an exponential function. That would be a function of the form $f(x)=Ae^{x-B}+C$ for some real $A,B,C$. Such functions have $y=C$ for a horizontal asymptote, but have no vertical asymptote, so such a function cannot satisfy your desired conditions.

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The OP asked for an "exponential function". (Which isn't possible given his/her criteria.) –  Fly by Night Jan 23 '13 at 20:01
    
@FlybyNight: Thank you. I missed that requirement. –  Cameron Buie Jan 23 '13 at 22:20
    
What I really needed was a non-linear function. The solution you provided works for us! Thank You. I have added another problem to my original question. –  Connor Finn McKelvey Jan 23 '13 at 23:11
    
@Connor: Your other problem is, indeed, quite similar. We'll proceed in almost exactly the same way, but we'll drop the "$-$" sign from each of the $4$ intermediate steps (since we need a negative function), and make substitutions for each of the corresponding numbers (i.e.: $100$ should now be $18$ in $g_2$ and thereafter, $55$ should now be $8$, $1$ should be $-1$, etc.). –  Cameron Buie Jan 24 '13 at 1:22
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