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Difference between revisions of "2019 AMC 10B Problems/Problem 9"
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==Problem==
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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>?
 
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>?
  
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==Solution==
 
==Solution==
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There are 4 cases we need to test here:
 
There are 4 cases we need to test here:
  
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iron
 
iron
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==See Also==
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{{AMC10 box|year=2019|ab=B|num-b=8|num-a=10}}
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{{MAA Notice}}

Revision as of 14:40, 14 February 2019

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 $f$?

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

Solution

There are 4 cases we need to test here:

Case 1: x is a positive integer. WLOG, assume x=1. Then f(1) = 1 - 1 = $0$.

Case 2: x is a positive fraction. WLOG, assume x=0.5. Then f(0.5) = 0 - 0 = $0$.

Case 3: x is a negative integer. WLOG, assume x=-1. Then f(-1) = 1 - 1 = $0$.

Case 4: x is a negative fraction. WLOG, assume x=-0.5. Then f(-0.5) = 0 - 1 = $-1$.

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

iron

See Also

2019 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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