Difference between revisions of "2019 AMC 10B Problems/Problem 9"

Revision as of 00:46, 19 October 2020

Problem

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 $f$ ?

$\textbf{(A) } \{-1, 0\} \qquad\textbf{(B) } \text{The set of nonpositive integers} \qquad\textbf{(C) } \{-1, 0, 1\} \qquad\textbf{(D) } \{0\}$
$\textbf{(E) } \text{The set of nonnegative integers}$

Solution 1

There are four cases we need to consider here.

Case 1: $x$ is a positive integer. Without loss of generality, assume $x=1$ . Then $f(1) = 1 - 1 = 0$ .

Case 2: $x$ is a positive fraction. Without loss of generality, assume $x=\frac{1}{2}$ . Then $f\left(\frac{1}{2}\right) = 0 - 0 = 0$ .

Case 3: $x$ is a negative integer. Without loss of generality, assume $x=-1$ . Then $f(-1) = 1 - 1 = 0$ .

Case 4: $x$ is a negative fraction. Without loss of generality, assume $x=-\frac{1}{2}$ . Then $f\left(-\frac{1}{2}\right) = 0 - 1 = -1$ .

Thus the range of the function $f$ is $\boxed{\textbf{(A) } \{-1, 0\}}$ .

~IronicNinja, edited by someone else hehe

Solution 2

It is easily verified that when $x$ is an integer, $f(x)$ is zero. We therefore need only to consider the case when $x$ is not an integer.

When $x$ is positive, $\lfloor x\rfloor \geq 0$ , so $\begin{split}f(x)&=\lfloor|x|\rfloor-|\lfloor x\rfloor| \\ &=\lfloor x\rfloor-\lfloor x\rfloor \\ &=0\end{split}$

When $x$ is negative, let $x=-a-b$ be composed of integer part $a$ and fractional part $b$ (both $\geq 0$ ): $\begin{split}f(x)&=\lfloor|-a-b|\rfloor-|\lfloor -a-b\rfloor| \\ &=\lfloor a+b\rfloor-|-a-1| \\ &=a-(a+1)=-1\end{split}$

Thus, the range of f is $\boxed{\textbf{(A) } \{-1, 0\}}$ .

Note: One could solve the case of $x$ as a negative non-integer in this way: $\begin{split}f(x)&=\lfloor|x|\rfloor-|\lfloor x\rfloor| \\ &=\lfloor -x\rfloor-|-\lfloor -x\rfloor-1| \\ &=\lfloor -x\rfloor-(\lfloor -x\rfloor+1) = -1\end{split}$

Video Solution

https://youtu.be/PgqjsTkNYdc

~savannahsolver

@@ Line 1: / Line 1: @@
-==Problem==
+== Problem ==
 The function <math>f</math> is defined by <cmath>f(x) = \lfloor|x|\rfloor - |\lfloor x \rfloor|</cmath>for all real numbers <math>x</math>, where <math>\lfloor r \rfloor</math> denotes the greatest integer less than or equal to the real number <math>r</math>. What is the range of <math>f</math>?
 <math>\textbf{(A) } \{-1, 0\} \qquad\textbf{(B) } \text{The set of nonpositive integers} \qquad\textbf{(C) } \{-1, 0, 1\} \qquad\textbf{(D) } \{0\}</math> <br> <math>\textbf{(E) } \text{The set of nonnegative integers} </math>
-==Solution 1==
+=== Solution 1 ===
 There are four cases we need to consider here.
@@ Line 21: / Line 19: @@
 ~IronicNinja, edited by someone else hehe
-==Solution 2==
+=== Solution 2 ===
 It is easily verified that when <math>x</math> is an integer, <math>f(x)</math> is zero. We therefore need only to consider the case when <math>x</math> is not an integer.
@@ Line 42: / Line 39: @@
 &=\lfloor -x\rfloor-(\lfloor -x\rfloor+1) = -1\end{split}</cmath>
-==Video Solution==
+=== Video Solution ===
 https://youtu.be/PgqjsTkNYdc
 ~savannahsolver
-==See Also==
+== See Also ==
 {{AMC10 box|year=2019|ab=B|num-b=8|num-a=10}}
 {{MAA Notice}}

2019 AMC 10B (Problems • Answer Key • Resources)
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