 ### 10.1 General information on the notation of functions

Designations:
 $ℝ$ Set of real numbers ${ℝ}^{+}$ Set of positive real numbers $M\subseteq N$ $M$ is a subset of $N$ $M\subset N$ $M$ is a subset of $N$ different from $N$ $𝔻$ Definition set $𝕎$ Value set $f:𝔻\to 𝕎$ The function $f$ maps the definition set $𝔻$ to the value set $𝕎$.

In the economic sciences, functions $f:𝔻\to 𝕎$ of the following types frequently occur: Definition: A function $f:𝔻\to 𝕎$ is a

1. real function, if $𝔻\subseteq ℝ$ and $𝕎\subseteq ℝ$
2. real-valued function (in $n$ real variables), if $𝔻\subseteq {ℝ}^{n}$ and $𝕎\subseteq ℝ$
3. ${R}^{n}$-${R}^{m}$-function, if $𝔻\subseteq {ℝ}^{n}$ and $𝕎\subseteq {ℝ}^{m}$
Examples:
1. $f:𝔻\to 𝕎,x↦{x}^{2}$ with $𝔻=ℝ$: We insert any real number into the function term $f\left(x\right)={x}^{2}$:
 $f\left(2\right)={2}^{2}=4,f\left(-7\right)={\left(-7\right)}^{2}=49,f\left(\sqrt{3}\right)={\left(\sqrt{3}\right)}^{2}=3,f\left(0\right)={0}^{2}=0$

Thus, you get the number $0$ or positive real numbers as function values. The value set $𝕎$ results to be $𝕎={ℝ}_{0}^{+}$. Thus, you can also write

 $f:ℝ\to {ℝ}_{0}^{+},x↦{x}^{2}$

.

2. $f:𝔻\to 𝕎,\left(x,y\right)↦-{x}^{2}-{y}^{4}$ with $𝔻={ℝ}^{2}$: Here any pairs $\left(x,y\right)\in {ℝ}^{2}$ are inserted:
 $f\left(2,1\right)=-{2}^{2}-{1}^{4}=-5,f\left(-1,-2\right)=-{\left(-1\right)}^{2}-{\left(-2\right)}^{4}=-1-16=-17,f\left(0,0\right)=-{0}^{2}-{0}^{4}=0$

Here we get as function values the number 0 or negative real numbers. The value set $𝕎$ results to be $𝕎={ℝ}_{0}^{-}$. Thus, you can also write

 $f:{ℝ}^{2}\to {ℝ}_{0}^{-},\left(x,y\right)↦-{x}^{2}-{y}^{4}$

.

3. $f:{ℝ}^{3}\to {ℝ}^{2},\phantom{\rule{0.3em}{0ex}}\left(\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill {x}_{2}\hfill \\ \hfill {x}_{3}\hfill \\ \hfill \hfill \end{array}\right)↦\left(\begin{array}{c}\hfill {x}_{1}{x}_{2}+{x}_{3}^{2}\hfill \\ \hfill {x}_{1}+{x}_{2}+{x}_{3}\hfill \\ \hfill \hfill \end{array}\right)$

(c) by Christian Bauer
Prof. Dr. Christian Bauer
Chair of monetary economics
Trier University
D-54296 Trier
Tel.: +49 (0)651/201-2743
E-mail: Bauer@uni-trier.de
URL: https://www.cbauer.de