Question

what are the even and odd components of 3rect(t+0.5)+rect(t-0.5)?
PS. rect is the rectangle function

Answer 1

whats rect?

Answer 2

Sometimes, you just need to look at it a bit. Copy this to your browser.

http://www.wolframalpha.com/input/?i=3*rect(t%2B0.5)%2Brect(t-0.5)&x=0&y=0

Answer 3

i think it's all even? no odd components since if you shift the rectangles you get even functions?

Answer 4

The standard definition for \(\mathrm{rect}(t)\) is
\[\mathrm {rect}(t)=\begin{cases}1&\text{for }|t|\le\dfrac{1}{2}\\[1ex]
0&\text{for }|t|>\dfrac{1}{2}\end{cases}\]Scaling by \(k\) and including a shift in the argument by \(c\) results in
\[k~\mathrm {rect}(t-c)=\begin{cases}k&\text{for }\dfrac{2c-1}{2}\le t\le\dfrac{2c+1}{2}\\[1ex]
0&\text{for }t<\dfrac{2c-1}{2}~{\huge\land}~ t>\dfrac{2c+1}{2}\end{cases}\]This means
\[3~\mathrm{rect}\left(t+\frac{1}{2}\right)+\mathrm {rect}\left(t-\frac{1}{2}\right)=\begin{cases}
(3\times1)+(1\times0)=3&\text{for }-1\le t< 0\\[1ex]
(3\times1)+(1\times1)=4&\text{for }t=0\\[1ex]
(3\times0)+(1\times1)=1&\text{for }0< t\le 1\\[1ex]
0&\text{for }|t|>1
\end{cases}\]So to answer your question, it depends on what you mean by "even and odd components". Is that asking when the value of the function is even/odd? or which parts show symmetry?