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For example, a line that :

  1. from -infinity to 0: y = 50
  2. from 200 to +infinity: y = 100
  3. and between 0 and 200, the linear connecting line (y=1/4x+50)

Thanks!

House3272
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1 Answers1

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Yes. You almost literally just did exactly that. This is typically denoted using a big curly brace containing the different cases like this (or similarly):

$$f(x) = \begin{cases} 50,& x < 50\\ 100,& x>100 \\ \frac{1}{4}x+50,& \text{else}\end{cases}$$

Though you may think this does not classify as "simple". In that case you can get the same by using indicator functions. It just looks different notation-wise:

$$f(x) = 50\cdot 1_{(-\infty, 50)}(x) + 100\cdot 1_{(200,\infty)}(x) +\left(\frac{1}{4}x+50\right)\cdot 1_{[50,200]}(x)$$

or perhaps:

$$f(x) = 50 + \frac{1}{4}x\cdot 1_{[50,200]}(x) + 50\cdot 1_{(200,\infty)}(x) $$

Stefan Perko
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  • I prefer Iverson brackets instead of indicator functions. Easier to read and write :) – Masacroso Sep 30 '16 at 22:29
  • Hmmmm, is there anyway to do this without explicitly specifying the points? Not the best example, but sort of like how x^3 has two 'parts'? – House3272 Sep 30 '16 at 22:30
  • No, not as a power series e.g., because this function is not differentiable at $50$ and $200$. $x^3$ being a polynomial in particular is differentiable everywhere. – Stefan Perko Sep 30 '16 at 23:28
  • @Masacroso Though there are advantages thinking of functions as opposed to expressions. – Stefan Perko Sep 30 '16 at 23:33