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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 \(x \ge 0\) for \(0 \le x \le 10.\)

 

I've solved part (a), and here's my answer:

 

In order for \(f(x)\) to equal \(1\), we must have the inequality \(1 \leq x \lfloor x \rfloor <2\). From this we can easily see that if \(x<1\), then \(\lfloor x \rfloor < 1\).

 

Similarly, if \(x\geq2\), then \(\lfloor x \rfloor\geq2\).

 

However, if \(1\le x<2\), then \(x\lfloor x \rfloor=x\), and so \(f(x)=1\). Hence, we have our answer.

 Jun 22, 2022
 #1
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Do not post A o P S homework.

 Jun 22, 2022

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