Let f(x)=⌊x⌊x⌋⌋ for x≥0
(a) Find all x≥0 such that f(x) = 1.
(b) Find all x≥0 such that f(x) = 3.
(c) Find all x≥0 such that f(x) = 5.
(d) Find the number of possible values of f(x) for 0≤x≤10.
thank you so much
+118703 Let's see
I'm going to look at c
f(x)=⌊x⌊x⌋⌋=5find all x≥0⌊x⌊x⌋⌋=55≤x⌊x⌋<65≤x⌊x⌋<6letx=k+δwhere k is an integer k≥0 and 0≤δ<15≤k(k+δ)<6x=k+δ=2.5works, in fact x can equal any number from 2.5 up to and not including 32.5≤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
