I experiencing difficulty understanding how to determine the impulse response for a given function. I think I understand it intuitively, that is the impulse response is a unit impulse function applied to a given system e.g. \[ h(t) = x(t) \delta (t) \] .

This is where my confusion is:

If \[ x(t) = e^{-\alpha*t}\: u(t) \] then if the unit impulse is applied then \[ h(t) = \delta (t)(e^{-\alpha*t}\: u(t)) = u(t) \]

I can understand that but then I see an example where \[ x(t) = u(t) \] and its impulse response is \[ h(t) = e^{-\alpha*t}\: u(t) \]

How is this possible, I am missing a vital piece of this jigsaw. Could someone please advice.