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I would like a formula for a function whose graph has the following properties:

  1. $f(0) = 0$.
  2. $\lim\limits_{x\to\infty}f(x) = y$.
  3. The shape of the function is approximately the following:

    http://i.imgur.com/oH876.png

  4. It should have an exponential or a logarithm in the formula.

Any function like this?

  • 1
    Well, yes: the function whose value at $x$ is the height of the graph you have is such a function.2012-05-26
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    I think what Arturo is hinting at is that in mathematical terminology, what you want is *formula*, not a *function*. @Arturo, there are clearer ways to make that point, don't you think?2012-05-26
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    @ArturoMagidin Could you give any function expression? Thanks!2012-05-26
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    @Rahul: Actually, no, I did not mean that, because I was not aware that this is what the OP is trying to do. Hard enough to figure out what he means by what he writes.2012-05-26
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    @RahulNarain Yes, I need a formula.2012-05-26
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    @Arturo: Yes, I noticed that just after I commented. Thanks.2012-05-26
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    There are $\vert \mathbb{R}^\mathbb{R} \vert$ functions from $\mathbb{R}$ to itself. How many of those do you think have cutesy formulas?2012-05-26

3 Answers 3

-1

Also your graph doesn't have anything shown for x<0, so presumably its equal to zero there?

i.e. Use a piecewise defined function:

f(x) = a-a*exp(-x) : for positive x and f(x)=0 for non-positive x.

4

The graph of $y=5-5e^{-x}$ has the desired characteristics. More generally, if you want $\lim\limits_{x\to\infty}f(x)=a>0$, the function $f(x)=a-ae^{-x}$ works.

2

$$y = 5 - 5 \exp(-\alpha x)$$ where $\alpha >0$ will do the job. You can control the rate of growth by playing around with $\alpha$.