#### Chapter 6 Elementary Functions

Section 6.3 Power Functions

##### Example 6.3.1

If an object falling in the homogeneous gravitational field of the Earth is observed, then the following relation between the falling time and the travelled distance can be found:
 Falling time $t$ in seconds $0$ $\sqrt{\frac{2}{g}}$ $\sqrt{\frac{2}{g}}·1.5$ $\sqrt{\frac{2}{g}}·2$ $\sqrt{\frac{2}{g}}·3$ Travelled distance $s$ in metres $0$ $1$ $2.25$ $4$ $9$

Here, $g\approx 9.81\frac{\mathrm{m}}{\mathrm{s}{}^{2}}$ is the physical constant of the gravitational acceleration. Now, plotting these values in a diagram with the horizontal axis $s$ and the vertical axis $t$ results in the figure below.

This suggests that the relation between $t$ and $s$ can be described mathematically by the function

$t:\mathrm{ }\left\{\begin{array}{ccc}\hfill \left[0;\infty \right)& \hfill \to \hfill & ℝ\hfill \\ \hfill s& \hfill ⟼\hfill & \sqrt{\frac{2}{g}}·\sqrt{s}\hfill \end{array}$

with $s$ being the independent variable. This is a function, in whose mapping rule a root (more specifically, a square root) of the independent variable occurs. Then, the graph of this function contains the measurement points listed above:

This example shows that functions with mapping rules that contain roots of the independent variables occur naturally in applications of mathematics.
For natural numbers $n\in ℕ$, $n>1$, the functions

${f}_{n}:\mathrm{ }\left\{\begin{array}{ccc}\hfill {D}_{{f}_{n}}& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & \sqrt[n]{x}={x}^{\frac{1}{n}}\hfill \end{array}$

are called radical functions. Obviously, these include the square root ${f}_{2}\left(x\right)=\sqrt{x}$, the cube root ${f}_{3}\left(x\right)=\sqrt[3]{x}$, the fourth root ${f}_{4}\left(x\right)=\sqrt[4]{x}$, etc., as mapping rules of functions (see exponent rules).
##### Exercise 6.3.2
Transform the mapping rule of the radical functions using exponent rules such that only exponents still occur in the mapping rule.

##### Exercise 6.3.3
What is the function ${f}_{n}$ with $n=1$?

Of great relevance is now the maximum domain ${D}_{{f}_{n}}$ that a radical function can have. Obviously, it depends on the exponent $n$ of the root which values of $x$ are allowed to be inserted in the mapping rule to obtain real values as a result. So we see that the square root $\sqrt{ }$ has a real value as a result only for a non-negative number. However, if we consider the cube root $\sqrt[3]{ }$, then we see that the cube root has a real value as a result for all real numbers, for example, $\sqrt[3]{-27}=-3$. Generally, we have:
##### Info 6.3.4

${f}_{n}:\mathrm{ }\left\{\begin{array}{ccc}\hfill {D}_{{f}_{n}}& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & \sqrt[n]{x}\hfill \end{array}$

with $n\in ℕ$ and $n>1$ have the maximum domain ${D}_{{f}_{n}}=\left[0;\infty \right)$ if $n$ is even and the maximum domain ${D}_{{f}_{n}}=ℝ$ if $n$ is odd.

Thus, the graphs of the first four radical functions, ${f}_{2}$, ${f}_{3}$, ${f}_{4}$, ${f}_{5}$, look like as in the figure below.

From the graphs, it can be seen that all radical functions are strictly increasing.
##### Exercise 6.3.5
${f}_{n}:\mathrm{ }\left\{\begin{array}{ccc}\hfill {D}_{{f}_{n}}& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & \sqrt[n]{x}\hfill \end{array}$
with $n\in ℕ$, $n>1$, find the range ${W}_{{f}_{n}}$ depending on whether $n$ is even or odd.