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Find the domain of function f(x) = 2 log (10−x/10 +x). Determine whether the function is even or odd.

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Kate200468idn
[tex] f(x) = 2 log ( \frac{10-x}{10+x} )\\ \\D:\ \frac{10-x}{10+x}>0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \wedge\ \ \ \ \ \ \ \ \ \ \ \ 10+x \neq 0\\\\.\ \ \ \ \ (10-x)(10+x)>0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x \neq 0\\\\.\ \ \ \ \ x\in (-10;10)\\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D=(-10;10)-\{0\}\\------------------------------ \\ [/tex]

[tex]the\ function\ is\ even\ \ \Leftrightarrow\ \ \ f(-x)=f(x)\ \ \ \wedge\ \ \ x,(-x)\in D\\\\the\ function\ is\ odd\ \ \ \Leftrightarrow\ \ \ f(-x)=-f(x)\ \ \ \wedge\ \ \ x,(-x)\in D\\-----------------------------\\\\f(-x)=2log( \frac{10-(-x)}{10+(-x)} )=2log( \frac{10+x}{10-x} )=2log( \frac{10-x}{10+x} )^{-1}=-2log( \frac{10-x}{10+x} )=\\ \\=-f(x)\ \ \ \ \Rightarrow\ \ \ the\ function\ is\ odd[/tex]
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