Mathematics

Problem

Define

k: \mathbb R \to \mathbb R

k(x)=\lfloor x \rfloor + |x-2|

, where

\lfloor x \rfloor

denotes the greatest integer less than or equal to

x

. (i) Obtain an explicit algebraic expression for

k(x)

when

1.5\le x<2

, when

2\le x<3

and when

3\le x<3.5

. (ii) Solve completely the equation

k(x)=3

, giving the full set of pre-images of

3

. (iii) Investigate whether there exists a real number

y

such that the equation

k(x)=y

has no real solution for

x

. If such a

y

exists, state it with justification; if not, explain why. a. a.