#### Chapter 6 Elementary Functions

Section 6.2 Linear Functions and Polynomials

# 6.2.6 Monomials

In addition to the linear affine functions studied in the previous section, we can also think of functions that assign to every real number a non-negative integer power of the number. An example is the function

$g:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & {x}^{2} .\hfill \end{array}$

This works for every non-negative integer exponent, and generally this function is written as

$f:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & {x}^{n}\hfill \end{array}$

with $n\in ℕ{}_{0}$, and it is called a monomial. The exponent $n$ of a monomial is called the degree of the monomial. For example, the function $g$ described at the beginning of this section is a monomial of degree $2$.
##### Exercise 6.2.10
Which functions are the monomials of degree $1$ and $0$?
The monomial of degree $2$ is called the standard parabola. The monomial of degree $3$ is called the cubic standard parabola. The figure below shows the graphs of a few monomials.

On the basis of these graphs, we now summarise some conclusions on monomials: There is a fundamental difference between monomials (with the mapping rule $f\left(x\right)={x}^{n}$, $n\in ℕ$) of even and odd degree. The range of monomials of an even non-zero degree is always the set $\left[0;\infty \right)$, while monomials of odd degree have the range $ℝ$. Furthermore, we always have

$f\left(1\right)={1}^{n}=1 ,$

$f\left(0\right)={0}^{n}=0$

and

Moreover, we have

$\left\{\begin{array}{cc}\hfill x>{x}^{2}>{x}^{3}>{x}^{4}>\dots & \text{for} x\in \left(0;1\right)\hfill \\ \hfill x<{x}^{2}<{x}^{3}<{x}^{4}<\dots & \text{for} x\in \left(1;\infty \right) .\hfill \end{array}$

##### Exercise 6.2.11
How can our conclusions concerning monomials be seen immediately from the exponent rules?