Power or polynomial function?

Question

According to the definition, $f(x) = a·x^n$ is a power function. If we shift it to $f(x) = a·(x - c)^n$ or, more general, to $f(x) = a·(x - c)^n + d$, it becomes a polynomial function (not a power function anymore). Is this just a matter of mere formalism or nomenclature?

Answer 1

Power functions must have a single term, but are allowed to have fractional or negative exponent.

Polynomials can have more than one term ("poly", meaning "many", refers to exactly this), but all exponents must be natural numbers.

A single term with a natural number exponent is both a power function and a polynomial (and actually a monomial) simultaneously.

Answer 2

In the natural languages, we often use the word "is" ambiguously, which I think is in the root of the confusion here.

When you say "the sky is blue" it may mean "the sky is one of blue things" but it may also mean "the 'colour of the sky' is blue".

The difference is subtle: in the first interpretation, you allow the sky to be of different colours at the same time, and we merely say that blue is one of those colours; in the second interpretation we assume that we can assign a unique "colour of the sky", and that the one assigned is blue.

Or, in mathematical language, in the first case you have a general relation of the objects to the colours, and in the second case it is assumed that this relation is a function (i.e. never one-to-many).

I presume that the natural language has grown that way because (a) In way too many cases it does not matter, because the relation happens to be the function, i.e. the element in relation happens to be unique, and (b) this itself is probably the result of evolution of language, as it reflects humans' natural tendency to organise and classify things into separate, disjoint, categories. I am speculating here, as I am not a language expert.

Still, in mathematics those two meanings are different and need to be carefully disambiguated. (And even in real life the sky can be blue-green with a shade of orange.)

Now to your point. When we say "$f$ is a power function" or "$f$ is a polynomial function", please take this as a statement akin to the first interpretation. We don't imply existence of a map which will unambiguously map $f$ to a label "constant function", "power function", "polynomial function", "exponential function", "trigonometric function" etc. A function can be many of these simultaneously. What would you say about the function $f(x)=0$? Notice that $f(x)$ is also $0\times x^5=e^{0\times x}-1=\sin^2x+\cos^2x-1$ etc.