#### Chapter 6 Elementary Functions

**Section 6.1 Basics of Functions**

# 6.1.1 Introduction

From Module 1 we already know that real numbers are sets and intervals are subsets of real numbers.

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For doing and applying mathematics, it is not sufficient to just consider sets and equations and inequalities for numbers of these sets, as done in the previous modules (for example, in Module 1). We also need functions (which are often also denoted as maps).

##### **Example 6.1.1 **

All real numbers $\mathbb{R}$, excluding the number $0\in \mathbb{R}$, are to be collected in a set. How is this set of numbers described? For this, the notation

$\mathbb{R}\setminus \{0\}$

is used. This reads as "$\mathbb{R}$ without $0$". Alternatively, this set can be described as a union of two open intervals:

$\mathbb{R}\setminus \{0\}=(-\infty ;0)\cup (0;\infty )\hspace{0.5em}.$

In the same way, single numbers can be removed from any other sets. So, for example, the set

$[1;3)\setminus \{2\}$

contains all numbers of the half-open interval $[1;3)$, excluding the number $2$:

is used. This reads as "$\mathbb{R}$ without $0$". Alternatively, this set can be described as a union of two open intervals:

In the same way, single numbers can be removed from any other sets. So, for example, the set

contains all numbers of the half-open interval $[1;3)$, excluding the number $2$:

##### **Exercise 6.1.2 **

Indicate the intervals $(-\infty ;\pi )$ and $(8;8.5]$ on the number line.

For doing and applying mathematics, it is not sufficient to just consider sets and equations and inequalities for numbers of these sets, as done in the previous modules (for example, in Module 1). We also need functions (which are often also denoted as maps).