Can someone show me how to show that:
f(t) = (t, 0 < t < 2
(4-t, 2 < t < 4
(0, t > 4
is equal to
f(t) = t - 2(t-2)U(t-2) + (t-4)U(t-4)
and determine its Laplace transform.
The way to think of this problem is that the Heaviside step functions (the U(t-a)'s) allow you to "switch on" certain portions of the function at different times. So, let's take your candidate. When t<2, both step functions are zero, because the arguments to both step functions are negative. You're left with f(t) = -4-t for t<2. That does not match up with the definition in the OP. So this can't be correct.
Technically, the function in the OP is undefined for t<0, so you don't really need a step function U(t) multiplying the initial t.
Keep in mind that the step functions are cumulative. As t marches past the various cutoffs, more and more step functions will be active. So, for example, when 2<t<4, you've got the initial t, but also the -2(t-2) multiplying the U(t-2). That coefficient of the step function is carefully chosen not only to cancel out the first piece (because it's still active), but also to get the desired 4-t.
Does this help?
As for the Laplace Transform, I would just use its linearity and the various properties and theorems.