f(t) = 10 e^{-t} 2[1/2 - u(t-1)], t>0,

so:

Lf = L[10e^{-t}] - 20 L[u(t-1)e^{-t}]

Lf(s) = 10/(s+1) - 20 e^{-s}/(s+1)

You have:Q2) f(t) = 2 [u(t) –u(t – 4)]+ t u(t – 4)

= 2u(t) – 2 u(t – 4)+ t u(t – 4)

= 2u(t)+ ( t – 2 ) u(t – 4) <----- how to derive to this step, it was addition and now ts multiplication. explaination and guide pls.

f(t) = 2u(t) – 2 u(t – 4)+ t u(t – 4)

now you take out the common factor u(t-4) from the last two terms on the right:

f(t) = 2u(t) + u(t – 4)(-2+ t)

rearrange and you have your mysterious line.

RonL