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

Solution:

If $x$ was an integer, then $\begin{align*}\lfloor|x|\rfloor - |\lfloor x \rfloor| &= |x| -|x|\\&=0.\end{align*}$

If $x$ was positive, then $\begin{align*}\lfloor|x|\rfloor - |\lfloor x \rfloor| &= \lfloor x \rfloor - \lfloor x \rfloor \\&= 0.\end{align*}$

Now, we must look at negative non-integers. If we have a negative non-integer, then $\lfloor|x|\rfloor$ would negate $x$ and then round down, while $|\lfloor x \rfloor|$ would round down then negate it, effectively negating it and rounding up.

The first one rounds down and the second one rounds up, the second one is $1$ larger than the first, making $f=-1.$

Therefore, the range is $\{-1,0\}.$

Thus, the answer is A .

2019 AMC 10B Problem 9

Solution:

Problem 9 in Other Years