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For all polynomial functions with odd order, the domain is all real numbers (meaning you can put any real number in the function) and the image is also all real numbers (meaning the function outputs every real number). Also, some power functions, such as $x^{2,2}$, have the same property.

However, none of elementary analytic functions that I can remember possess such qualities. For example, the image of $e^x$ doesn't contain negative numbers and $\sin x$ is limited to values between $-1$ and $1$. Likewise, the domain of $\ln x$ doesn't include negative values.

Then, what are some examples of non-polynomial (nor power) analytic functions whose domains and images both are the entirety of real numbers?

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    $\begingroup$ If you just want a "function", and don't require continuity, an obvious example is f(x)= 1/x for x not equal to 0, f(0)= 0. $\endgroup$
    – user247327
    Commented May 31, 2021 at 0:35
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    $\begingroup$ $f(x)=\frac{x^3}{x^2+1}$ $\endgroup$
    – MasB
    Commented May 31, 2021 at 0:38
  • $\begingroup$ Ah yes, quotients of polynomials! I had omitted them because the denominater could be zero so the function would be undefined for some $x$, but if its roots are imaginary, like in your example, we have no such issues. Thanks! $\endgroup$
    – MrPillow
    Commented May 31, 2021 at 0:45
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    $\begingroup$ Well, $\sinh x=\frac12(e^x-e^{-x})$ will do you nicely. Or, maybe more simply $x+e^x$. $\endgroup$
    – Lubin
    Commented May 31, 2021 at 1:01

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Conway's base-$13$ function provides an extreme example: its domain is $\mathbb{R}$ and for every nontrivial interval $(a,b)$ its restriction to that interval is surjective onto $\mathbb{R}$. Beyond merely not being a polynomial, this function is extremely discontinuous - its original motivation is as a counterexample to the converse of the intermediate value theorem.

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