CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

Let R be the set of real numbers and f:RR be defined by f(x)={x}1+[x]2, where [x] is the greatest integer less than or equal to x,  and {x}=x[x]. Which of the following statements are true?

I. The range of f is a closed interval.
II. f is continuous on R.
III. f is one-one on R
  1. I only
  2. II only
  3. III only
  4. None of I, II and III


Solution

The correct option is D None of I, II and III
f(x)={x}1+[x]2=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪x+12;    1x<0x;             0x<1x12;        1x<2x25;        2x<3...so on

For Maximum of f(x)={x} should be maximum and  [x] should be minimum 
maximum {x}1  
So range of f is [0,1) which is open interval.

Checking continuity at  x=0
L.H.L.
=limx0f(x)=12
R.H.L.
=limx0+f(x)=0
L.H.L R.H.L
f is not continuous at x=0 and hence, not continuous on R.

For any integer xI,f(x)=0.
So, it is not one-one function. 

flag
 Suggest corrections
thumbs-up
 
0 Upvotes


Similar questions
View More



footer-image