10.1 General information on the notation of functions

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 $𝕎\subseteq {ℝ}^m$

1. $f:𝔻\to 𝕎,x\mapsto x^2$ 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

   $$f:ℝ\to {ℝ}_0^{+},x\mapsto x^2$$

   .

2. $f:𝔻\to 𝕎,\left(x,y\right)\mapsto -x^2-y^4$ 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

   $$f:{ℝ}^2\to {ℝ}_0^{-},\left(x,y\right)\mapsto -x^2-y^4$$

   .

3. $f:{ℝ}^3\to {ℝ}^2,\left(\begin{array}{c}\hfill x_1\hfill \\ \hfill x_2\hfill \\ \hfill x_3\hfill \\ \hfill \hfill \end{array}\right)\mapsto \left(\begin{array}{c}\hfill x_1{x}_2+x_3^2\hfill \\ \hfill x_1+x_2+x_3\hfill \\ \hfill \hfill \end{array}\right)$