Domainomania-2

Algebra Level 3

$\large f(x)=\frac{1}{\lfloor |x-1| \rfloor + \lfloor |7-x|\rfloor-6}$

Find the sum of all integers for which the above function is undefined.

Notations:

$| \cdot |$ denotes the absolute value function .
$\lfloor \cdot \rfloor$ denotes the floor function .

7 6 28 9 10 11 12 8

2 solutions

Nishant Rai
Apr 22, 2015

Domain of given function can be represented as: $\Re -(0,1]\cup \{1,2,3,4,5,6,7 \}\cup [7,8)$

Deepak Kumar
Apr 22, 2015

It is obvious that we need to exclude 'x' for which denominator equals 0. Make cases corresponding to the change in definition of |x-1| and |7-x| .Case1: 1<x<7 => [|x-1|]+[|7-x|]=6=>[x-1]+[7-x]=6 => [x]-1+[-x]+7=6=> [x]+[-x] =0 => x cannot be integer => x lies in (1,7)-{2,3,4,5,6}. Similarly, work on two more cases like x<=1 & x>=7 .Remember [x+I]=x+I (Where I is an integer ) & [x]+[-x]=0 if x is integer otherwise [x]+[-x]=-1 .Also [x]=I => x lies in [I,I+1)

@Gautam Sharma Though the question is awesome ... the options are not up to mark. The required option is sum of $9$ variables ... it is easy to see that most probably the answer is $36$ .

Nihar Mahajan - 6 years, 1 month ago

Domainomania-2

2 solutions

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