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One question from the IYMC 2023 was

Find the roots of the function $$f(x) = \left(\pi^x-\frac{1}{\pi}\right)\cdot\left(x^\pi\right)\cdot\left(\frac{1}{\pi^2}-\pi^x\right)$$

Looking at it, I set each factor to $0$ and found $-1, -2$ and $0$ as roots. However, when I plot the graph in Desmos, the only root is $0$, and the domain of $x$ gets limited to the reals greater than or equal to $0$. When I try $f(-2)$, Desmos returns "undefined", but when I try plug $-2$ into the formula in my calculator, I get $0$.

Is this just an error on Desmos?

Blue
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Chris Daniel
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  • What's $(-2)^\pi$? It can be defined sort of meaningfully (in complex analysis), but that 1) goes well beyond the scope of the high-school math in the problem set on their website, and 2) it will be multivalued. (E.g. see Marc van Leeuwen's comment here). Overall the question is more attention checking than a competition problem, even on entry level. – Amateur_Algebraist Nov 11 '23 at 12:33
  • The value of $(-2)^\pi$ does not matter as the $\frac1{\pi^2}-\pi^{-2}=0$ will cancel it out – Тyma Gaidash Nov 11 '23 at 14:55

1 Answers1

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If you do it with $3$ instead of $\pi$ you get enter image description here

but with $3.1$ it's

enter image description here

John Hunter
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  • They just declare non-integer exponents for negative numbers to be undefined, since they work in the reals. Desmos is mainly concerned with plots; how would you plot such a function outside the supposed three roots? – Amateur_Algebraist Nov 11 '23 at 12:36
  • @ Amateur_Algebraist good point, perhaps Desmos is doing it's best – John Hunter Nov 11 '23 at 12:40