How to convert this piece-wise function to unit step?

Question

How would I approach this piece-wise function to convert it to unit step?

$$g(t) = \begin{cases}2t & 0 \leq t<1\\2 & 1\leq t < ∞\end{cases} $$

How do I start this? I'm not getting the second piece.

Answer 1

Hint: Each interval $I_{(a,b)}(t)$ can be written using step function $u(t)$: \begin{equation} u(t) = \begin{cases} 1, & 0\leq t <\infty\\ 0,& -\infty <t <0 \end{cases} \text{ and } I_{(a,b)}(t) = \begin{cases} 1, & a\leq t <b\\ 0,& \text{otherwise} \end{cases} \implies I(t) = u(t-a)-u(t-b). \end{equation}

Answer 2

Using these notes as a guide, in general, if

$$f(t)= \left\{\begin{array} fg(t) &0 \le t < a\\h(t) &a \le t < \infty \end{array}\right.$$

then

$$f(t) = g(t)u(t) - g(t)u(t-a) + h(t) u(t-a)$$

For your particular problem, we have:

$$f(t) = \begin{cases}2t & 0 \leq t<1\\2 & 1\leq t < ∞\end{cases}$$

Using unit step functions, we can write this as:

$$f(t) = 2tu(t) - 2t u(t-1) + 2 u(t-1)$$

Verify that each variant of $f(t)$ produces the following plot