What does the u(t) mean within sign in signals?

Question

I understand that within signals and math, u(t) is used to represent the unit step function. What I am confused about is the notation and the meaning behind the way that some functions are written. When for example, I have a signal which is $y(t) = \frac{2}{3}e^{-3t}u(t)$ what exactly does the u(t) mean in the function and how would this affect the signal if it wasn't written?

Answer 1

As $u(t)$ equals $1$ for $t\geq0$ and $0$ for $t < 0$, multiplying some function $f(t)$ by $u(t)$ produces new function $y(t) = f(t) u(t)$ that equals $f(t)$ for $t \geq0$ and $0$ for $t<0$. This in some way ensures that $y(t)$ is defined “nicely” for negative values of $t$, as having negative values of time variable in signal processing is not that common.