This piecewise function is not defined on the intervals (-oo,1) and [8,oo], but wolfram alpha is saying it has a domain of all Reals.
Why is this?
The plot even shows h(1/2) = 0. The function doesn't specify what happens at t=1/2, so how do they get that h(1/2)=0?
h[t] == Piecewise[{{8 - 3 t, 1 <= t <= 3}, {t - 4, 3 < t < 8}}]
http://www.wolframalpha.com/input/?i=domain+of+h%28t%29%3DPiecewise%28%28%288-3t%2C1%3C%3Dt%3C%3D3%29%2C%28t-4%2C3%3Ct%3C8%29%29%29
