Let g(x) = [x-1] + [1-x], x ε [-3,3]. Write g(x) as a piecewise function without greatest integer brackets.
would someone give me a brief explanation on how to solve such equations?
(note: these are greatest integer brackets)
let
f(x)=[x-1]
h(x)=[1-x] $\displaystyle x\in [-3,3] $
$\displaystyle f(x)=\begin{Bmatrix}
-4&...if...-3\leq x<-2\\
-3&...if...-2\leq x<-1\\
-2&...if...-1\leq x<0\\
-1&...if...0\leq x<1\\
0&...if...1\leq x<2\\
1&...if...2\leq x<3\\
2&...if...x=3
\end{Bmatrix}
$
$\displaystyle h(x)=\begin{Bmatrix}
4&...if...x=-3\\
3&...if...-3< x\leq -2\\
2&...if...-2< x\leq -1\\
1&...if...-1< x\leq 0\\
0&...if...0< x\leq 1\\
-1&...if...1< x\leq 2\\
-2&...if...2< x\leq 3 \\
\end{Bmatrix}$
$\displaystyle g(x)=\begin{Bmatrix}
0&...if...x=-3\\
-1&...if...-3< x< -2\\
0&...if...x=-2\\
-1&...if...-2< x< -1\\
0&...if...x=-1\\
-1&...if...-1< x< 0\\
0&...if...x=0\\
\end{Bmatrix}$
$\displaystyle
............\begin{Bmatrix}
-1&...if...0< x< 1\\
0&...if...x=1\\
-1&...if...1< x< 2\\
0&...if...x=2\\
-1&...if...2< x<3\\
0&...if...x=3\\
\end{Bmatrix}$ since we can write f(x) and h(x) like below then find the sum of f(x) and h(x) because g(x)=f(x)+h(x)
$\displaystyle f(x)=\begin{Bmatrix}
-4&...if... x=-3\\
-4&...if...-3< x<-2\\
-3&...if...x=-2\\
-3&...if...-2< x<-1\\
-2&...if...x=-1\\
-2&...if...-1< x<0\\
-1&...if...x=0\\
\end{Bmatrix}$
$\displaystyle
...........\begin{Bmatrix}
-1&...if...0< x<1\\
0&...if...x=1\\
0&...if...1< x<2\\
1&...if...x=2\\
1&...if...2< x<3\\
2&...if...x=3
\end{Bmatrix}
$
$\displaystyle h(x)=\begin{Bmatrix}
4&...if... x=-3\\
3&...if...-3< x<-2\\
3&...if...x=-2\\
2&...if...-2< x<-1\\
2&...if...x=-1\\
1&...if...-1< x<0\\
1&...if...x=0\\
\end{Bmatrix}$
$\displaystyle
...........\begin{Bmatrix}
0&...if...0< x<1\\
0&...if...x=1\\
-1&...if...1< x<2\\
-1&...if...x=2\\
-2&...if...2< x<3\\
-2&...if...x=3
\end{Bmatrix}$
it is clear or not