a$)$ the function the and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function.

b$)$ the function that assigns the next largest integer to a positive integer

c$)$ the function that assigns to a bit string the number of one bits in the string

d$)$ the function that assigns to a bit string the number of bits in the string

$5.$ Find the domain and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function.

a$)$ the function that assigns to each bit string the number of ones in the string minus the number of zeros in the string

b$)$ the function that assigns to each bit string twice the number of zeros in that string

c$)$ the function that assigns the number of bits left over when a bit string is split into bytes $($which are blocks of $8$ bits$)$

d$)$ the function that assigns to each positive integer the largest perfect square not exceeding this integer