Is $x^x$ a polynomial, an exponential or both?

Question

If $c$ is a constant, and $x$ is a variable, we'd say that $f(x) = x^c$ is a polynomial function of order $c$. Conversely, the function $f(x) = c^x$ would be called an exponential function.

Is there a name for a function of the form $f(x) = x^x$? Strictly speaking it's neither an exponential nor a polynomial.

Answer 1

$$x^x=e^{\ln x^x}=e^{x \ln x}$$

Therefore, it is a composition of an exponential and the product of $x \cdot \ln x$

Answer 2

It's neither. A poynomial is a function that is of the form $\sum_i c_ix^i$ where the $c_i$ are constants. An exponential function is one of the form $Ca^x$ for some constant $a$ and nonzero constant $C$ Note that $x$ is not a constant, and so $x^x$ is of neither form.

Answer 3

No, this is not a polynomial as a polynomial necessarily has non-negative integer exponents. I've heard it referred to as a "hyperpower" function and you can read about it, and similar topics, on this page.

Also, you could think of this function as $e^{x\log x}$ which is a composition of polynomial, exponential and logarithmic functions.