#### Chapter 6 Elementary Functions

Section 6.1 Basics of Functions

# 6.1.3 Functions in Mathematics and Applications

Mathematical functions often describe relations between quantities that arise from other sciences or everyday life. For example, the volume $V$ of a cube depends on the side length $a$ of the cube. The volume can be considered as a mathematical function that assigns the corresponding volume $V\left(a\right)={a}^{3}$ to every possible side length $a>0$:

$V:\mathrm{ }\left\{\begin{array}{ccc}\hfill \left(0;\infty \right)& \hfill \to \hfill & ℝ\hfill \\ \hfill a& \hfill ⟼\hfill & V\left(a\right)={a}^{3} .\hfill \end{array}$

The result is the cubic standard parabola (see Section 6.2.6) for the relation between side length and volume. In this way, many more examples can be found arising from sciences and everyday life: the position as a function of time in physics, the reaction rate as a function of concentration in chemistry, the amount of flour needed as a function of the desired amount of dough in a cake recipe, etc.
To this end, let us consider an example.
##### Example 6.1.12
The intensity of nuclear radiation is inversely proportional to the square of the distance to the source. This is called the inverse square law. Using a physical proportionality factor $c>0$, the relation between intensity $I$ of the radiation and distance $r>0$ from the source can be formulated mathematically as follows:

$I:\mathrm{ }\left\{\begin{array}{ccc}\hfill \left(0;\infty \right)& \hfill \to \hfill & ℝ\hfill \\ \hfill r& \hfill ⟼\hfill & \frac{c}{{r}^{2}} .\hfill \end{array}$

Hence, for the intensity we have the mapping rule $I\left(r\right)=\frac{c}{{r}^{2}}$ that describes the relation between the quantities $I$ and $r$.

##### Exercise 6.1.13
In the construction of wind turbines it is known that the wind turbine power is proportional to the cube of the wind velocity. Under the condition that the proportionality factor satisfies the relation $\rho >0$, which of the following mathematical functions describes this relation of physical quantities correctly?
a)

$P:\mathrm{ }\left\{\begin{array}{ccc}\hfill \left(0;\infty \right)& \hfill \to \hfill & ℝ\hfill \\ \hfill v& \hfill ⟼\hfill & P\left(v\right)=\frac{\rho }{{v}^{3}}\hfill \end{array}$

b)

$P:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & ℝ\hfill \\ \hfill v& \hfill ⟼\hfill & P\left(v\right)=\rho {v}^{3}\hfill \end{array}$

c)

$P:\mathrm{ }\left\{\begin{array}{ccc}\hfill \left[0;\infty \right)& \hfill \to \hfill & ℝ\hfill \\ \hfill v& \hfill ⟼\hfill & P\left(v\right)=\rho {v}^{3}\hfill \end{array}$

d)

$x:\mathrm{ }\left\{\begin{array}{ccc}\hfill \left[0;\infty \right)& \hfill \to \hfill & ℝ\hfill \\ \hfill f& \hfill ⟼\hfill & x\left(f\right)=\rho {f}^{3}\hfill \end{array}$