Let be defined by .

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

Problem

Let

f:\mathbb{R} \to \mathbb{R}

be defined by

f(x)=2\lfloor x \rfloor+|x|

. (i) For an integer

n

, write an explicit expression for

f(x)

when

n \le x < n+1

. Explain your reasoning. (ii) Evaluate

f(-2.4)

f(-0.2)

f(3)

and

f(4.7)

. (iii) Determine all real numbers

x

for which

f(x)=5

. Give your answer as a union of intervals or a set of points, as appropriate.