When I run into algebraic patterns in my engineering / physics modeling multiple times, I usually go see what the smart mathematicians have learned about that, and often discover something useful that improves my modeling. For example, the $w e^w$ pattern… once you realize that’s the “Lambert W” function, then you learn all kinds of very useful stuff.
However, my searches for this pattern haven’t turned up anything. Maybe it's not really a common pattern. So, I figured this group would know one way or the other… do any of these functions (or generalizations) have names:
$$\begin{align} f(x) &= x^x \\ f(x, a) &= x^{a x} \\ f(x, a, b) &= x^{a x + b} \\ f(x, a, b, c) &= x^{a x^2 + b x + c} \\ &= x^{\operatorname{quadratic}(x, a, b, c)} \end{align}$$
If not, if you were a mathematician capturing the properties of these, what would you name these? (What search terms might turn up properties and/or typical uses?)
(Or if you were providing some optimized code to compute these in your modeling, what would you name these?)