Differences between types of exponential functions?

Question

Can someone point out the key differences between functions of the form $ax^n$ and $ab^x$?

I'm only able to see certain differences when I plot examples for both, for example, it looks like $ax^n$ will either be an even function or a function with the domain as $\mathbb{R}$.

More?

Answer 1

The key difference that it should be pointed out is that

$ax^n$ is not an exponential function, it's a polynomial. While $ab^x$ is indeed an exponential function.

We call a function exponential when the indipendent variable appears as the exponent of some number. We instead call polynomial every function consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables (cfr. Wikipedia).

We know lots of properties about this two type of function. As an example here's a simple list

1. Exponentials are either increasing ($b>1$) or decreasing ($0<b<1$) or constant ($b=1$) $\forall x \in \mathbb R$ On the other hand, polynomials are not. They can increase or decrease constantly in some intervals (e.g. the parabola $y=x^2$ is decreasing for $x\in(-\infty, 0]$ and increasing for $x\in[0, \infty)$)
2. Exponentials are always positive or negative, depending on the constant $a$: $ab^x\gt 0$ $\forall x\in\mathbb R$ if $a>0$, if $a<0$ then $ab^x\lt0$ $\forall x$. On the other hand polynomials are not always one ore another.

And there are many more properties, more than I could write down in hours. For completeness see the relative pages on wikipedia: exponential funcion, polynomials.