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 \(0 \le x \le 10.\)

 Apr 25, 2020

I have not seen a question like this before but this looks kind of like a step broken parabola to me.

As x increases so does f(x) but it happens in steps. 


part a)

If x=1 then f(1)=1 

If x=2 then f(2)=4

If x=1.99 then f(x) = floor of 1.99*1 = 1  this will be true of any x value between 1 and 2


for   \(1\le x<2\qquad f(x)=1\)



for  \(0\le x<1\qquad f(x)=0\)


I have not really through much further than that but that gives you a good start for your own thought process. 

Please be careful, it is tricky.  Try drawing the graph to help you.

I'm going for a walk. You can show me what you have done when I get back if you want to. I'll be happy to help more.

 Apr 25, 2020

thank you!

littlemixfan  Apr 25, 2020

could you help me with part d? 

littlemixfan  Apr 25, 2020

What answers do you have for b and c first.

OR what have you done to try and solve them?

I do not want to do all the work without you demonstrating that you are learning.

Melody  Apr 25, 2020

Try these


\(f(0)=\\ f(1)=\\ f(2)=\\ f(2\frac{1}{2})=\\ f(3)=\\ f(3\frac{1}{3})=\\ f(3\frac{2}{3})=\\ f(4)=\\ f(4\frac{1}{4})=\\ f(4\frac{2}{4})=\\ f(4\frac{3}{4})=\\ f(5)=\\ f(5\frac{1}{5})=\\ etc\)


Try some values in between.

Can you see what is happening?

 Apr 25, 2020

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