Define the function p(x)= |x-4| - x

MATHEMATICS · Multi-part · AoPS · Mathematics · Special Functions · Special Functions

Problem

Define the function

p(x)=\sqrt{|x-4| - \lfloor x \rfloor},

where

\lfloor x \rfloor

denotes the greatest integer less than or equal to

x

. (i) Analyse the expression

|x-4|

on the intervals

(-\infty,4]

and

[4,\infty)

. (ii) For an arbitrary integer

n

, consider

x\in[n,n+1)

. Express the radicand

|x-4|-\lfloor x \rfloor

in terms of

n

and

x

, and state the condition under which it is non-negative. (iii) Using part (ii), determine all real numbers

x

for which the radicand is non-negative, and hence write the domain of

p(x)

in interval notation.