The sets A$,$ B and C are defined as follows:

A $=\{0,1,2,3,4,5,6,7\}$

B $=\{-8,-6,-4,-2,0,2,4,6,8,10\}$

C $=\{7,11,13,17\}$

$($a$)$ In this question you will define functions by specifying the output for each

potential input. There is no need to create a mathematical expression or formula

for your functions.

$($i$)$ Define a function f from A to A which is both one$-$to$-$one and onto.

Explain how you know it is one$-$to$-$one and onto.

$($ii$)$ Define a function g from A to B$.$ Decide whether it is one$-$to$-$one or onto,

explaining your answer.

$($iii$)$ Define a function h from B to C$.$ Decide whether it is one$-$to$-$one or

onto, explaining your answer.

$($b$)$ Using the functions f$,$g$,$ and h from $($a$),$ find the values of the following

expressions, showing all the intermediate steps you took to get your answers.

$($i$)$ h$($g$($f$(3)))[1]$

$($ii$)$ g$($f$-1(6))[1]$

$($iii$)$ f$-1($f $-1(5))[1]$

$($c$)$ Let $=\{5,1,0,4\}.$ Define a $($non$-$trivial$)$ function : using an

algebraic expression. Your function should include at least one squaring

operation involving x and at least one subtraction by a number. For example,

$()=(1)2$ or $()=21.$

$**$ Bonus marks are available in this question if your expression contains a

$($non$-$trivial$)$ division as well as satisfying the other conditions.

You should not choose an example from lectures or workshops.

$($i$)$ For the given domain, what is the range of $?$

$($ii$)$ Derive an algebraic expression for

$1().$

$($iii$)$ Find the maximal domain and codomain for so that both and

$1$

are well defined. Explain your answer.