Should a hash function be reversible?

Question $1$ Answer

a$.$

No$,$ because this would allow the plaintext to be decrypted from the hash, rendering message digests and digital signatures unworkable.

b$.$

Yes, because this would allow the hash to contribute to error detection and correction operations.

c$.$

In a limited way, they are reversible, because the hash acts as a pointer or key into database and file management systems where the plaintext comes from.

d$.$

No$,$ because a hash is a many$-$to$-$one function and thus must have a collision detection and avoidance mechanism as part of its implementation; being reversible would negate this.