$2.$ Laplace transform and its inverse

a$)$ Consider f $($t$)=$

$$

$$

$$

$$

$$

$0$

$5$

t $+1$

$1$

$0<$ t $<3$

$3<$ t $<4.5$

$4.5<$ t $<6$

t $>6$

$.$

i$)$ Write f $($t$)$ in terms of step functions and t$.(6$ points$)$

ii$)$ Find Laplace transform of f $($t$).(6$ points$)$

b$)$ Find time domain function f $($t$)$ for the s$-$domain function F$($s$)=$ s $+1$

$0.1$s$2+$ s $+2.9.(8$

points$)$