Prirodna domena funkcije je najveći podskup realnih brojeva na kojem je funkcija dobro definirana. U ovom lancu zadataka svaki se skup može prikazati kao konačna unija disjunktnih intervala, a njegovu duljinu ćemo definirati kao zbroj duljina spomenutih intervala (duljinu intervala definiramo kao razliku njegove gornje i donje granice). Primjerice duljina skupa
![\{17,18\} = [17,17]\cup [18,18]](/media/m/f/e/9/fe949b8b64ec62b8bbcc831e1e45cba6.png)
je
![17-17+18-18=0](/media/m/2/3/d/23d226bc272aa293cfdb278e7a8081cc.png)
, dok je duljina skupa
![[-3, 5\rangle](/media/m/0/3/2/032aef631a32a23297cef6fd12c44b0b.png)
jednaka
![5+3=8](/media/m/e/6/8/e684b5b24ba551b8ece7864d8927c4fb.png)
. Duljina praznog skupa je
![0](/media/m/c/f/9/cf91351cd8c1329cb9841cca01ec0763.png)
i neće se pojavljivati duljine neograničenih skupova. Odredi duljinu prirodne domene funkcije
Natural domain of a function is the largest subset of real numbers on which the function is well defined. In this chain of problems every set can be represented as a finite union of disjoint intervals, and we define its length as a sum of the lengths of the mentioned intervals (length of an interval is a difference between its upper and lower bounds). For example the length of the set
![\{17,18\} = [17,17]\cup [18,18]](/media/m/f/e/9/fe949b8b64ec62b8bbcc831e1e45cba6.png)
is
![17-17+18-18=0](/media/m/2/3/d/23d226bc272aa293cfdb278e7a8081cc.png)
, while the length of the set
![[-3,5 \rangle](/media/m/8/5/e/85e139f23e01c4d9b0896b9c2f0fc3e6.png)
equals
![5+3=8](/media/m/e/6/8/e684b5b24ba551b8ece7864d8927c4fb.png)
. Length of an empty set is
![0](/media/m/c/f/9/cf91351cd8c1329cb9841cca01ec0763.png)
and there won't be unbounded sets. Determine the length of the natural domain of the function
[lang=hr]
Prirodna domena funkcije je najveći podskup realnih brojeva na kojem je funkcija dobro definirana. U ovom lancu zadataka svaki se skup može prikazati kao konačna unija disjunktnih intervala, a njegovu duljinu ćemo definirati kao zbroj duljina spomenutih intervala (duljinu intervala definiramo kao razliku njegove gornje i donje granice). Primjerice duljina skupa \(\{17,18\} = [17,17]\cup [18,18]\) je \(17-17+18-18=0\), dok je duljina skupa \([-3, 5\rangle\) jednaka \(5+3=8\). Duljina praznog skupa je \(0\) i neće se pojavljivati duljine neograničenih skupova. Odredi duljinu prirodne domene funkcije
\[f(x)=\sqrt{(x^2-2)\left( \frac{1}{x^2}-2\right) }.\]
%sqrt(2)=1.414213562
[/lang]
[lang=en]
Natural domain of a function is the largest subset of real numbers on which the function is well defined. In this chain of problems every set can be represented as a finite union of disjoint intervals, and we define its length as a sum of the lengths of the mentioned intervals (length of an interval is a difference between its upper and lower bounds). For example the length of the set \(\{17,18\} = [17,17]\cup [18,18]\) is \(17-17+18-18=0\), while the length of the set \([-3,5 \rangle\) equals \(5+3=8\). Length of an empty set is \(0\) and there won't be unbounded sets. Determine the length of the natural domain of the function
\[f(x)=\sqrt{(x^2-2)\left( \frac{1}{x^2}-2\right) }.\]
[/lang]