2019 AMC 10B Problem 9

Question

The function

f

is defined by

f(x) = \lfloor|x|\rfloor - |\lfloor x \rfloor|

for all real numbers

x,

where

\lfloor r \rfloor

denotes the greatest integer less than or equal to the real number

r.

What is the range of

f?

\{-1, 0\}

The set of nonpositive integers

\{-1, 0, 1\}

\{0\}

The set of nonnegative integers

Check positive numbers, integers, and negative non-integers separately

For negative non-integers, compare rounding before and after absolute value

If

x\ge0

, then

\lfloor|x|\rfloor=\lfloor x\rfloor=|\lfloor x\rfloor|

, so

f(x)=0

.

If

x

is a negative integer, both terms equal

|x|

, so again

f(x)=0

.

If

x

is negative and not an integer, write

x=-k-t

, where

k

is a nonnegative integer and

0<t<1

. Then

\lfloor|x|\rfloor=k

, while

|\lfloor x\rfloor|=|-k-1|=k+1

, so

f(x)=-1

.

Therefore, the range is

\{-1,0\}.

Thus, the answer is A .

Problem 9 in other years