$(20$ pts$)$ Observe the following information to answer the following set of questions:

$f$:$\{0,1{\}}^4\{0,1{\}}^4$ and $f(x)$ is obtained by replacing the first bit of $x$ with $1.$ For example, $f(0100)=1100.$

$g$:$\{0,1{\}}^4\{0,1{\}}^4$ and $g(x)$ is obtained by removing the first bit of and placing the bit at the end. For example, $g(1000)=0001.$

a$.$ Show the domain and co$-$domain for both $f($on left$)$ and $g($on right$)($map the functions$).$

b$.$ Show the range for both functions and determine the cardinality for each.

c$.$ Discuss if the functions are well$-$formed $($why or why not$).$

d$.$ For both functions $f$ and $g,$ show whether they are injective.

e$.$ For functions $f$ and $g,$ show if they are surjective.

f$.$ Based on the above, describe using logic formation if the functions are bijective.