$(2\mathrm{.}5$ marks$)$

Consider the function $,f$:$R_{x}R$

$x(x-7{)}^2-4$

$($a$)$ What is the domain of definition of f$?$ Justify your answer.

$($b$)$ Explain why the range of $f$ is not $R.$

$($c$)$ What is $($or are$)$ the image$($s$)$ of $0?$ Justify your answer.

$($d$)$ What is $($or are$)$ the preimage$($s$)$ of $0?$ Justify your answer.

$($e$)$ Assume the codomain of f is $R^{**}$ instead of $R.$ What is then the domain of definition of f $?$

Justify your answer.

$(2\mathrm{.}5$ marks$)$

Same questions as above when

$f$:$R$

$xR$

$x|\sqrt[\phantom{2}]{x-7}-1||$

$(2\mathrm{.}5$ marks$)$

Same questions as above when $,f$:$RR$

$x|4+\frac{1}{(x-7{)}^2}||$

$(2\mathrm{.}5$ marks$)$

Same questions as above when $f$:$RR$

$x|4-|x-7|||$