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\begin{align} f(t) = \begin{cases} 3t &\mbox{if } t \leq 3 \\ 12 & \mbox{if } 3

I'm confused as to how I can write this in terms of the unit step function with the first f(t) has $\leq 3$ instead of just $\lt 3$. Any help with this problem?

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    This is besides the question, but your function is slightly flawed. At $t=7$, is $f(t)$ equal to 12 or 0?2012-12-14

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Depends on the definition of the Heaviside function - it can be either 0, 1 or 0.5 at H(0).

If you take it to be zero, I think the following works:

[3tH(3-t) + 12 H(7-t) H(t-3)]H(7-t)

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    Indeed, http://www.wolframalpha.com/input/?i=%5B3t+H%283-t%29+%2B+12+H%287-t%29+H%28t-3%29%5DH%287-t%292012-12-14