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Let $p(x)$ be defined on $2 \le x \le 10$ such that$$p(x) = \begin{cases} x + 1 &\quad \lfloor x \rfloor\text{ is prime} \\ p(y) + (x + 1 - \lfloor x \rfloor) &\quad \text{otherwise} \end{cases}$$where $y$ is the greatest prime factor of $\lfloor x\rfloor.$ Express the range of $p$ in interval notation.

I did some work on this problem: Desmos Graph

And I put $[3,5) \cup [6,6] \cup [8,8]$. What did I do wrong?

  • Did you mistype your result? The intervals $[6,6]$ and $[8,8]$ look strange. (And they are incorrect.) – Daniel Fischer Sep 01 '20 at 21:20
  • There was another thread that said that if one singular point is in the interval, then that is how you would do it. How would you do it? Also, does everything else seem right? –  Sep 01 '20 at 21:37
  • I would usually write ${6}$ rather than $[6,6]$. But sometimes the interval notation may be the more natural choice. Okay, so you really meant singleton intervals. But if you correctly wrote the definition of $p$ those are wrong. What reasoning led you to them? – Daniel Fischer Sep 01 '20 at 21:42
  • What do you mean by "But if you correctly wrote the definition of $p$ those are wrong." What did I do wrong again? –  Sep 01 '20 at 21:50
  • I can't tell you what you did wrong unless I know what you did. I have a hypothesis what you did wrong (misread the first clause of the definition), but unless you explain your reasoning I don't know if my hypothesis is correct. – Daniel Fischer Sep 01 '20 at 21:55
  • My reasoning was that every number in the range $2 \leq x < 3$, $3 \leq x < 4$, all the way to $9 \leq x <10$ and ultimately $10$ produces either one single number or a line. I just plugged in values from here. –  Sep 01 '20 at 23:00
  • Every number produces only a single number. Which number is for example $p(5.5)$? – Daniel Fischer Sep 02 '20 at 11:52
  • Well, $p(5.5)=6$, but for example for $x=6$, we have $3+(x+1-6)=x-2$ for $6 \leq x < 7$, implying that every segment of length 1 either produces a line segment of a single number. –  Sep 02 '20 at 21:51
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    According to the definition in the question, $p(5.5) = 6.5$. Does that help you see the mistake? – Daniel Fischer Sep 02 '20 at 21:53
  • Ahh, thank you! I forgot about that. Thanks! –  Sep 02 '20 at 21:55

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