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Mathematics-Online course: Prepcourse Mathematics - Analysis - Functions

Function

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A function

$\displaystyle f: D\rightarrow \mathbb{R}\,,\quad x \mapsto f(x) $

associates each argument $ x$ in the domain $ D \subseteq \mathbb{R}$ to a value $ f(x)$ in the range $ W \subseteq\mathbb{R}$ .

The graph of $ f$ consists of all pairs $ (x,y)$ with $ y=f(x)$ .

\includegraphics[bb=140 533 337 718,clip,width=.45\linewidth]{Def_Bereich}

As the illustration shows, domain (colored in light-grey) and range (colored in dark-grey) simply are the graph's projections onto the $ x$ - and $ y$ -axis respectively.

(Authors: Höllig/Hörner/Knödler/Abele)
In order to determine domain and range of the function

$\displaystyle f(x)= \frac{\ln(3-x)}{\sqrt{x}-1},$

start with the restrictions given by the involved elementary functions. So, the logarithm's argument has to be positive while the square root can only be extracted from non-negative arguments:

$\displaystyle 3-x > 0 \,\,\,\,$   und$\displaystyle \,\,\,\, x \geq 0.$

Also, the denominator must not be zero:

$\displaystyle x\neq 1\,.$

In total this means:

$\displaystyle \mathrm{D}=[0,3) \backslash \{1\} = [0,1)\cup(1,3).$

The range of $ f$ becomes obvious by drawing its graph.

\includegraphics[bb=114 558 339 717,clip,width=.45\linewidth]{Def_Bereich_Bsp}

It clearly is $ W=\mathbb{R}$, since for $ x \in (1,3)$, $ f$ takes all values between $ +\infty$ and $ -\infty$ .

(Authors: Höllig/Hörner/Knesch/Abele)
The following table shows domain $ D$ and range $ W$ of some elementary functions:

$ f(x)$ $ D$ $ W$
$ 1/x$ $ \mathbb{R}\setminus\{0\}$ $ \mathbb{R}\setminus\{0\}$
$ \ln x$ $ (0,\infty)$ $ \mathbb{R}$
$ \tan x$ $ \mathbb{R}\setminus\{x\,:\,x=(2k+1)\pi/2,\ k\in\mathbb{Z}\}$ $ \mathbb{R}$
$ \sqrt{x}$ $ [0,\infty)$ $ [0,\infty)$
(Authors: Höllig/Hörner/Knesch/Abele)
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  automatically generated 9/18/2007