Finding a single expression for a signal using unit-step functions

Question

So suppose that we have the signal x(t) shown in the figure below. How would I use unit-step functions to find just a single expression for x(t) that is valid for all t.

Unit Step Question Image

Answer 1

Let $u_a(t)$ is the unit step function which is $0$ for $t<a$ and $1$ for $t\geq a$. Then the function $u_a - u_b$ is $0$ for $t$ outside the interval $[a,b)$ and $1$ inside the interval. This little construction is the key to such things. If you want to turn $f(t)$ on over the interval $[a,b)$ then write $f(t)(u_a -u_b)$. You get $f(t)$ in the interval and $0$ outside. So for the function in your picture, you want $t+2$ between $-2\leq t <0$, and you write $(t+2)(u_{-2}-u_0).$ You want $2-t$ for $0 \leq t < 1$, so write $(2-t)(u_0-u_1)$. You want $1$ for $1\leq t <2$, so write $1(u_1-u_2)$. Put them all together and the function you want is

$$(t+2)(u_{-2}-u_0)+(2-t)(u_0-u_1)+(u_1-u_2).$$