para = {U ->  UnitStep, h ->  Exp[-s]}

Q 5.4 (1)   p.98

ft = 1 - t + (t - 1) U[t - 1] /. para
Plot[ft, {t, 0, 2}, AspectRatio ->  0.2]

Expand[LaplaceTransform[ft, t, s]]

Q 5.4 (2)  k=1   p.98

ft = U[t - 1] - U[t - 4] /. para
Plot[ft, {t, 0, 6}, AspectRatio ->  0.2]

LaplaceTransform[ft, t, s]

Q 5.4 (3)  k=1    p.98

ft = U[t] - U[t - c] /. para
Plot[ft /. c ->  1, {t, 0, 6}, 
      AspectRatio ->  0.2]

LaplaceTransform[ft, t, s]
% /. {UnitStep[-c] ->  0, 
        UnitStep[c] ->  1}

Q 5.4  (4)  p.98

ft = t - (t - k) U[t - k] /. para
Plot[ft /. k ->  1, {t, 0, 6}, AspectRatio ->  0.2, 
      PlotRange ->  {0, 2}]

Simplify[LaplaceTransform[ft, t, s]]
% /. {UnitStep[-k] ->  0, 
        UnitStep[k] ->  1}

Q 5.4 (5)  p.98

ft = 
      t - (t - k) U[t - k] - k U[t -  k] /. 
        para
Plot[ft /. k ->  1, {t, 0, 6}, 
      AspectRatio ->  0.2]

Simplify[LaplaceTransform[ft, t, s]]
% /. {UnitStep[-k] ->  0, 
        UnitStep[k] ->  1}

Q 5.4 (6)  p.98

ft = 
      2 - t + (t - 1) U[t - 1] - U[t - 1] /. 
        para
Plot[ft, {t, 0, 6}, AspectRatio ->  0.2, 
      PlotRange ->  {0, 2}]

Expand[Simplify[LaplaceTransform[ft, t, s]]]