Functions Question 372
Question: Find the domain of the function $ f(x)=\sqrt{( \frac{2}{x^{2}-x+1}-\frac{1}{x+1}-\frac{2x-1}{x^{3}+1} )} $
Options:
A) $ (-\infty ,2]-{-1} $
B) $ (-\infty ,2) $
C) $ ]-1,2] $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ f(x)=\sqrt{( \frac{2}{x^{2}-x+1}-\frac{1}{x+1}-\frac{2x-1}{x^{3}+1} )} $ We must have $ \frac{2}{x^{2}-x+1}-\frac{1}{x+1}-\frac{2x-1}{x^{3}+1}\ge 0 $ Or $ \frac{2(x+1)-(x^{2}-x+1)-(2x-1)}{(x+1)(x^{2}-x+1)}\ge 0 $ Or $ \frac{-(x^{2}-x-2)}{(x+1)(x^{2}-x+1)}\ge 0 $ Or $ \frac{-(x-2)(x+1)}{(x+1)(x^{2}-x+1)}\ge 0 $ Or $ \frac{2-x}{x^{2}-x+1}\ge 0, $ where $ x\ne -1 $ Or $ 2-x\ge 0,x\ne -1 $ (as $ x^{2}-x+1>0\forall x\in R $ ) Or $ x\le 2,x\ne -1 $ Hence. Domain of the function is $ (-\infty ,-1)\cup ( -1,2 ]. $ or $ ( -\infty ,2 ]-{-1} $
BETA
