0

I want to write a mathematical expression of this graph enter image description here

First thing that I notice is it's actually $f(t) = 6-4t$ with it domains restricted at $[0,2]$. Can I write expression of $x(t)$ as $x(t)=x(t+2)=6-4t$? Or is there any more "right" way to write the expression of $x(t)$?

Thank you

Why Would You
  • 461
  • 1
    You can write $x(t) = x(t+2)$, but it being equal to $6-4t$ only works on the domain you listed. – The Chaz 2.0 Oct 25 '20 at 05:14
  • 1
    The notation ${,x,}$ is sometimes used for the "fractional part" of $x$, that is, the number satisfying $0\le{,x,}<1$ and $x-{,x,}$ is an integer. Your function can be expressed simply in terms of ${,x,}$. Alternatively, it has an expression as an infinite sum $b_0+\sum_1^{\infty}(a_i\sin(\pi x)+b_i\cos(\pi x))$ for some constants $a_i,b_i$. – Gerry Myerson Oct 25 '20 at 05:48
  • 1
    It can be written as a sum of sinusoids. may be this will help you: https://kconrad.math.uconn.edu/math1132s10/sawtooth.html#:~:text=Fairly%20general%2C%20even%20discontinuous%2C%20periodic,3x)%20%2B%20....&text=sin(x)%20%2D%201%E2%81%84,(6x)%20%2B%20... – Moti Oct 25 '20 at 06:21
  • @Moti an infinite sum of sinusoids. – Gerry Myerson Oct 26 '20 at 00:37
  • @Gerry yes. This is one example how to use a family of periodic functions to depict a certain periodic function. The accuracy of the final function requirement will set the "number" of the periodic functions required. – Moti Oct 26 '20 at 15:07
  • So, Airlangga, have you followed up on any of the ideas in these comments? – Gerry Myerson Oct 27 '20 at 03:08
  • @GerryMyerson Yep! This problem has 2 questions. The first one is asking the mathematical expression of $x(t)$ and the other one is asking to express $x(t)$ using Fourier Series. So, for the first one, I think I'll write it as $x(t)=x(t+4)$. Thank you! – Why Would You Oct 27 '20 at 03:41
  • $x(t)=x(t+4)$ only says the function has period $4$. Lots of functions whose graphs don't look anything like yours satisfy that equation. [and, anyway, your function has period $2$, so $x(t)=x(t+2)$, as you know.] – Gerry Myerson Oct 27 '20 at 06:11

0 Answers0