$1.$ Laplace transform and its inverse $(10$ points$)$

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

$$

$$

$$

$$

$$

e$5$t

$0$

$($t $7)$

$0<$ t $<3$

$3<$ t $<5$

t $>5$

$.$

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

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

b$)$ i$)$ Find time domain function f $($t$)$ for the s$-$domain function F$($s$)=8$s

$4$s$2+12$s $+25.$

ii$)$ Find time domain function f$1($t$)$ for the s$-$domain function F$1($s$)=$ e$2$s F$($s$).$