f(x)= mod(x-1) + mod(x-2) + mod(x-3) FIND RANGE OF THIS?

f(x)= mod(x-1) + mod(x-2) + mod(x-3)                                     FIND RANGE OF THIS?



3 Answers

basit ali
41 Points
11 years ago

here , x-1≥0 


similarly, x≤2 andx≤3 (because this is a modulus function.)

by taking common through  sign - scheme it's range is(-∞,3].

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Ashwin Muralidharan IIT Madras
290 Points
11 years ago

Hi Srichandra,


This is a question on range of a modulus function, where you need to break it into intervals and solve it.


We can clearly break this into interval where there is a sign change in a particular modulus,

ie less than 1, between 1 and 2, between 2 and 3, and greater than 3.


Let x be in [3,infinity)

Which will give f(x) = y = (x-1) + (x-2) + (x-3) = 3x - 6, so range is [3,infinity) {when x = 3, y = 3 and will take all values greater than 3 when x is greater than 3}


Next x be in [2, 3)

Which will give y = (x-1) + (x-2) - (x-3) = x, so range is [2, 3). { same as x }


Next x be in [1,2)

Which will give y = (x-1) - (x-2) - (x-3) = 4-x, so range is (2,3]


Next x is less than 1, ie x be in (-infinity,1)

which will give y = -(x-1) -(x-2) -(x-3) = 6-3x, so range in thsi interval is (3,infinity).


Hence the range of this function is the union of the ranges in all the respective intervals, which would be the interval [2,infinity) which is the range of this function.

Note that the range has to be posititve in any case, since it is sum of modulus functions, each of which is individually positive.


You can also have questions on this like what would be the minimum value of this function, which in this case would be 2, when x = 2.


Hope that helps.


All the best,


Ashwin (IIT Madras).

ankit singh
askIITians Faculty 614 Points
2 years ago
x3=0 (Denominator cannot be zero)
Domain: R{3}
Range ={1,1}.

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