How to express a continuous signal with basic functions

Question

My goal is to express this continuous signal by using $u(t)$ (unit step) and $r(t)$ (unit ramp)basic functions

• $u(t)$

• $r(t)$

• The signal

So i would say that $x(t) = u(t) - r(t-1)$

Although my calculation is wrong, i have to add $+r(t-2)$ to $x(t)$ i can not quite understand why.

Why the correct answer is $x(t) = u(t) - r(t-1) + r(t-2)$ ?

Note: I'm new with with this kind of mathematics and expressions , any edit would be appriciated!

Answer 1

The signal vanishes after t = 2.

To add the upward function $r(t-2)$ is to compensate, or neutralize, the downward signal $-r(t-1)$ for t > 2. Otherwise, the signal would last forever.