Is $(1+x)^m$ a power function by definition?

Question

Definition: A one-term function with a variable as the base and a constant as the exponent. The exponent in the case of a power function is always a constant and a real number.

The general form of a power function is, $f(x)=kx^n$

I know the foregoing definition as the definition of power functions, but i encounter with a function such as $(1+x)^m$, it is also known as binomial function.The website says that $(1+x)^m$ is a power function, but i think that it does not obey the given definition.

So , can we say that $(1+x)^m$ is a power function, if so, can you explain please ?

Answer 1

Definitions

It depends on your definition. The definition given in the question is

Definition: A one-term function with a variable as the base and a constant as the exponent. The exponent in the case of a power function is always a constant and a real number.

The website in the question gives a similar definition:

Definition: A power function has the form $$ f(x) = ax^p$$ where:

• $a$ and $p$ are constants,
• $p$ is a real number,
• and $a$ is nonzero.

Thus, per these definitions, the only functions which are power functions are those which can be written in the form $ x \mapsto kx^m,$ where $k$ and $m$ are real constants. A function of the form $$ x \mapsto (1+x)^m$$ is not of this form, hence it is not, by definition, a power function. It is interesting to note that the cite website is, itself, not even consistent about its use of this definition, as their discussion of the binomial theorem starts with the text

The binomial series is a type of Maclaurin series for the power function $f(x) = (1 + x)^m$.

Note, however, that it is not necessary to define a power function in this way. We write definitions in order to clearly communicate ideas and to capture the essential properties of an object or idea we want to talk about. So, for example, when I teach this material in precalculus classes, I generally give the following definitions:

Definition: A primitive power function is a function of the form $$ p : (0,\infty) \to \mathbb{R} : x \mapsto x^m, $$ where $m$ is a non-zero real number.

It is helpful to study "primitive power functions", since they have similar properties: they are either monotonically increasing or decreasing (depending on the sign of $p$, their graphs come in only a couple of shapes (basically depending on where $p$ lives relative to $0$ and $1$), and so on.

A more general power function is then something of the form $$ x \mapsto b\left( \frac{x-h}{a} \right)^m + k, $$ which is just a primitive power function which has been scaled, translated, and (possibility) reflected. Per this definition, the function $$ x \mapsto (1+x)^m$$ is a power function.

Does it Matter?

In higher level mathematics, we are often kind of sloppy about terminology, particularly when it is not terribly important. I think that Tao has said some useful words on the "why?" of this. The notion of a "power function" is somewhat useful in calculus (since the power rule is a useful computational tool), but bare "power functions" don't generally pop up that often beyond the pedagogical setting of an introductory calculus class. Polynomial and exponential functions are much more common.

As such, one probably should not get too caught up on the precise definition of a "power function". Yes, the Statistics How To website is inconsistent in its usage of the term (which might belie some deeper problems with their material, but that's neither here nor there), but it largely doesn't matter, since that isn't really the core of what is being described and discussed.

Answer 2

As you have said, it doesn't look it fits the definition. I think the definition is a bit imprecise here, but if we decide we want to make the function fit the definition, we can rewrite it with a substitution.

Set $w=1+x.$ Then you get $$f(x)=(1+x)^m=w^m=f(w-1).$$ This now fits the criteria nicely.

Alternatively, you can skip a step and simply let the input be $x-1$ to get $$f(x-1)=(1+(x-1))^m=x^m.$$

As mentioned by @csch2, this is then a power function translated one unit to the left.