Functions Question 413
Question: The domain of the function $ f(x)=log_{e}{sgn(9-x^{2})}+\sqrt{{{[x]}^{3}}-4[x]} $ (where [.] represents the greatest integer function) is
Options:
A) $ [ -2,1 )\cup [ 2,3 ) $
B) $ [ -4,1 )\cup [ 2,3 ) $
C) $ [ 4,1 )\cup [ 2,3 ) $
D) $ [ 2,1 )\cup [ 2,3 ) $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] We have $ f(x)=log_{e}{sgn(9-x^{2})}+\sqrt{{{[x]}^{3}}-4[x]} $ We must have, sgn $ (9-x^{2})>0\Rightarrow 9-x^{2}>0 $
$ \Rightarrow x^{2}-9<0\Rightarrow (x-3)(x+3)<0\Rightarrow -3<x<3 $ (i) Also $ {{[x]}^{3}}-4[x]\ge 0\Rightarrow x\ge 0 $
$ \Rightarrow x([x]+2)\ge 0 $
$ \Rightarrow [x]\ge -2 $ or [x] lies between -2 and 0 i.e., $ [x]=-2,-1 $ or 0 Now, $ [x]\ge -2\Rightarrow x\ge 2 $ (ii) $ [x]=-2\Rightarrow -2\le x<-1;[x]=-1\Rightarrow -1\le x<0 $ $ [x]=0\Rightarrow 0\le x<1 $ Hence $ [x]=-2,-1,0 $
$ \Rightarrow -2\le x<1 $ Hence $ D_{f}=[-2,1]\cup [ 2,3 ). $
BETA
