Question

The graph of $f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|$ is symmetric about which of the following? (Here $\lfloor x \rfloor$ is the greatest integer not exceeding $x.$ )

the $y$ -axis

the line $x=1$

the origin

the point $\left(\dfrac{1}{2},0\right)$

the point $(1,0)$

Answer: D

Concepts:floor and ceiling functions symmetry (algebra)

Difficulty rating: 1540

Solution:

For every real $x,$ $\begin{aligned} &f(1-x)=|\lfloor 1-x\rfloor| \\ &\quad {}-|\lfloor x\rfloor|=-f(x). \end{aligned}$

Equivalently, replacing $x$ by $\frac{1}{2}+t$ gives $f\!\left(\frac{1}{2}-t\right)=-f\!\left(\frac{1}{2}+t\right).$ This is point symmetry about $\left(\frac{1}{2},0\right).$

Thus, D is the correct answer.

2021 AMC 10A Fall Problem 16

Answer: D

Difficulty rating: 1540

Solution:

Problem 16 in Other Years