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

Answer: A

Concepts:floor and ceiling functions absolute value casework

Difficulty rating: 1370

Solution:

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 .

2019 AMC 10B Problem 9

Answer: A

Difficulty rating: 1370

Solution:

Problem 9 in Other Years