$2\mathrm{.}2$ Let $H$ denote a system operator that maps a function to a function; let $x$ and $y$ denote functions; and let all other variables denote numbers. Using strictly$-$correct mathematical notation, write an expression for each quantity specified below. Only use brackets for grouping when strictly required. Use $D$ to denote the derivative operator.

$($a$)$ the output of the system $H$ when its input is $y$;

$($b$)$ the output of the system $H$ evaluated at $2t-1$ when the input to the system is $x$;

$($c$)$ the output of the system $H$ evaluated at $t$ when the input to the system is $ax$;

$($d$)$ the output of the system $H$ evaluated at $5t$ when the input to the system is $x+y$;

$($e$)$ the derivative of the output of the system $H$ when its input is $ax$;

$($f$)$ the output of the system $H$ when its input is the derivative of $ax$;

$($g$)$ the sum of: $1)$ the output of the system $H$ when its input is $x$; and $2)$ the output of the system $H$ when its input is $y$;

$($h$)$ the output of the system $H$ when its input is $x+y$; and

$($i$)$ the derivative of $x$ evaluated at $5t-3.$