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Let \(f(x) = \lfloor x \lfloor x \rfloor \rfloor\) for \(x \ge 0\)

(a) Find all \(x \ge 0\) such that f(x) = 1.

(b) Find all \(x \ge 0\) such that f(x) = 3.

(c) Find all \(x \ge 0\) such that f(x) = 5.

(d) Find the number of possible values of f(x) for \(0 \le x \le 10.\)

 

thank you so much

 May 2, 2022
 #1
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Remember to use that f(x) is related to the floor function!

 May 2, 2022
 #2
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Good idea, thanks!

Guest May 2, 2022
 #3
avatar+117116 
+1

Let's see

I'm going to look at c

 

\(f(x)=\lfloor x\lfloor x\rfloor \rfloor=5 \qquad \text{find all }x\ge 0\\ \lfloor x\lfloor x\rfloor \rfloor=5\\ 5\le x\lfloor x\rfloor <6\\ 5\le x\lfloor x\rfloor <6\\ let \;\;x=k+\delta\qquad \text{where k is an integer }k\ge0 \text{ and }0\le\delta<1\\ 5\le k(k+\delta )<6\\ x=k+\delta = 2.5\;\;works,\\ \text{ in fact x can equal any number from 2.5 up to and not including 3}\\ 2.5\le x <3 \)

 

 

 

 

LaTex:

f(x)=\lfloor x\lfloor x\rfloor \rfloor=5 \qquad \text{find all }x\ge 0\\
\lfloor x\lfloor x\rfloor \rfloor=5\\
5\le x\lfloor x\rfloor <6\\
5\le x\lfloor x\rfloor <6\\
let \;\;x=k+\delta\qquad \text{where k is an integer  }k\ge0    \text{ and  }0\le\delta<1\\
5\le k(k+\delta )<6\\
x=k+\delta = 2.5\;\;works,\\
\text{ in fact x can equal any number from 2.5 up to and not including 3}\\
2.5\le x <3

 May 2, 2022
 #4
avatar+117116 
+1

You should do, or at least seriously try,  the others for yourself.

 May 2, 2022

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