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I need a function that meets certain criterias.

This is what I've got so far:

(tan^-1(x*2-1.5)+1)/1.5

It looks like this:

graph

And it is nearly perfect, except that I want exactly:

  1. x = 1 --> y = 1

  2. x = 2 --> y = 1.5

  3. x = 0 --> y = 0

I can replace either of the 1.5's in the function above to satisfy either 1 or 2, but don't know how to find numbers that will satisfy both, though I have a hunch that it's some fancy number, involving PI.

so, I guess my question is: What function looks like the above graph and satisfies my three statements?

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There are no numbers $a, b$ so that the function $f(x) = \frac{\arctan(2x + a) + 1}{b}$ satisfies all three properties.

The condition $f(0) = 0$ implies $\arctan(a) + 1 = 0$, i.e. $$a = \tan(-1) \approx 1.56.$$ But this, combined with the condition $f(1) = 1$, implies $$b = \arctan(2 + \tan(-1)) + 1 \approx 1.42.$$ With these values for $a$ and $b$ $$f(2) \approx 1.54.$$

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