Question

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

Answer 1

`f(x) = 2 log ( (10-x)/(10+x) )\\ \\D:\ (10-x)/(10+x)>0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \wedge\ \ \ \ \ \ \ \ \ \ \ \ 10+x \\eq 0\\\\.\ \ \ \ \ (10-x)(10+x)>0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x \\eq 0\\\\.\ \ \ \ \ x\in (-10;10)\\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D=(-10;10)-\{0\}\\------------------------------ \\`


`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( (10-(-x))/(10+(-x)) )=2log( (10+x)/(10-x) )=2log( (10-x)/(10+x) )^(-1)=-2log( (10-x)/(10+x) )=\\ \\=-f(x)\ \ \ \ \Rightarrow\ \ \ the\ function\ is\ odd`