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This might seem like a stupid question, but can $$x^{-2x^2}$$ be called a polynomial function?

Thanks!

suomynonA
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asdf
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No, $x^{-2x^2}$ is not a polynomial function. the exponent on the variable in a polynomial function must be a constant natural number (non-negative integer). For example, $x^2$ is a polynomial function. On the other hand, $x^{-2}$ is a power function but not polynomial function because $-2$ is negative and so not a natural number. Then $x^x$ grows faster with $x$ than even an exponential function, so is certainly not polynomial. And $x^{-2x^2}$, which also has variable in both base and exponent, is even worse. It instead exhibits exponential decay, which polynomials do not.

ziggurism
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  • What do you mean by "faster than even an exponential"? – suomynonA Oct 13 '16 at 01:55
  • @suomynonA one property of real functions often discussed is their growth rates. Functions with exponentiation with a variable base and constant exponent are polynomial, whereas functions with exponentiation with constant base and variable exponent are called exponential. Exponentials of positive growth rates always outgrow any polynomial, no matter how large the exponent of the polynomial. Functions with variable base and variable exponent can have even larger growth rates. That means they go to infinity faster than exponentials. Polynomials cannot do this. – ziggurism Oct 13 '16 at 01:59
  • Oh, do you mean that $x \rightarrow ∞$, terms with variable exponents increase faster that those without variable exponents? I just couldn't understand your statement. – suomynonA Oct 13 '16 at 02:04
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    Let me edit for clarity – ziggurism Oct 13 '16 at 02:04
  • Just a small comment: the (sole) exponent can be zero (some definitions of "natural number" include this, some don't). In this case, you have a constant polynomial. More precisely, one can say "non-negative integer exponents". – Deepak Oct 13 '16 at 02:10
  • @Deepak you rather mean non-negative integers. – adjan Oct 13 '16 at 02:13
  • @adjan Yes, I thought I'd written it that way. Brain fart. Edited now.Thanks for the catch. – Deepak Oct 13 '16 at 02:14