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Version 1.2 von Tina Müller am 2024/10/15 08:36
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Tina Müller 1.1 1 Untersuche die Funktion //f// mit {{formula}}f(x)=\frac{1}{x}{{/formula}} und Definitionsbereich {{formula}}\mathbb{R}^*{{/formula}} im Hinblick auf ihr Randverhalten und ihre Wertemenge. Ergänze dafür zunächst folgende Wertetabellen.
2
3 (% style="list-style: alphastyle" %)
4 1. (((Randverhalten: Verhalten im Unendlichen
5 1) Verhalten gegen plus Unendlich ({{formula}}+\infty{{/formula}})
6 (% class="border" %)
7 |={{formula}}x{{/formula}}| {{formula}}+1{{/formula}}| {{formula}}+10{{/formula}}| {{formula}}+100{{/formula}}| {{formula}}+1000{{/formula}}| {{formula}}+10^6{{/formula}}| {{formula}}+10^9{{/formula}}| {{formula}}+10^{12}{{/formula}}|({{formula}}+10^{+\infty}{{/formula}})
8 |={{formula}}f(x){{/formula}}|1|{{formula}}\frac{1}{10}{{/formula}}|{{formula}}\frac{1}{100}{{/formula}}|{{formula}}\frac{1}{1000}{{/formula}}|{{formula}}\frac{1}{1000000}{{/formula}}|{{formula}}\frac{1}{1000000000}{{/formula}}|{{formula}}\frac{1}{1000000000000}{{/formula}}|0
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10 2) Verhalten gegen minus Unendlich ({{formula}}-\infty{{/formula}})
11 (% class="border" %)
12 |={{formula}}x{{/formula}}| {{formula}}-1{{/formula}}| {{formula}}-10{{/formula}}| {{formula}}-100{{/formula}}| {{formula}}-1000{{/formula}}| {{formula}}-10^6{{/formula}}| {{formula}}-10^9{{/formula}}|{{formula}}-10^{12}{{/formula}}|
Tina Müller 1.2 13 |={{formula}}f(x){{/formula}}||{{formula}}-1{{/formula}}|{{formula}}-\frac{1}{100}{{/formula}}|{{formula}}-\frac{1}{1000}{{/formula}}|{{formula}}-\frac{1}{1000000}{{/formula}}|{{formula}}\frac{1}{1000000000}{{/formula}}|{{formula}}\frac{1}{1000000000000}{{/formula}}|0
Tina Müller 1.1 14 )))
15 1. (((Randverhalten: Verhalten nahe der Definitionslücke ({{formula}}x \approx 0{{/formula}})
16 1) Verhalten links bei der Definitionslücke ({{formula}}x \approx 0{{/formula}} mit {{formula}}x<0{{/formula}})
17 (% class="border" %)
18 |={{formula}}x{{/formula}}| {{formula}}-1{{/formula}}| {{formula}}-0,1{{/formula}}| {{formula}}-0,01{{/formula}}| {{formula}}-0,001{{/formula}}| {{formula}}-10^{-6}{{/formula}}| {{formula}}-10^{-9}{{/formula}}| {{formula}}-10^{-12}{{/formula}}|0
19 |={{formula}}f(x){{/formula}}||||||||
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21 2) Verhalten rechts bei der Definitionslücke ({{formula}}x \approx 0{{/formula}} mit {{formula}}x>0{{/formula}})
22 (% class="border" %)
23 |={{formula}}x{{/formula}}| {{formula}}+1{{/formula}}| {{formula}}+0,1{{/formula}}| {{formula}}+0,01{{/formula}}| {{formula}}+0,001{{/formula}}| {{formula}}+10^{-6}{{/formula}}| {{formula}}+10^{-9}{{/formula}}| {{formula}}+10^{-12}{{/formula}}|0
24 |={{formula}}f(x){{/formula}}||||||||
25 )))
26 1. Erkennst du eine Symmetrie?
27 1. Beschreibe das Randverhalten der Funktion und nenne ihre Wertemenge.
28 1. Bestimme {{formula}}g(y){{/formula}} für {{formula}}y=g(x){{/formula}} und {{formula}}x\in \mathbb{R}^*{{/formula}}.