Domains and Ranges of sets in algebra

Question

State whether the following functions are well-defined. If they are, then give the Domain, Co-domain and Range and state whether the functions are one-to-one and/or onto and give your reasoning. If they are not well defined, then explain why not.

a) f: J → J, f(x) = 3x + 1

b) g: N → N, g(x) = x2 – 1

c) h: N → R, h(x) = +x

d) j: {words} → {letters}, j(x) = initial letter of x

I will appreciate if someone gives correct answers to each part. Thanks

Answer 1

This looks like homework, so I'll just guide you through the las problem and see if you can do the rest on your own.

d) j: {words} → {letters}, j(x) = initial letter of x

Domain: words Co-Domain: letters Range: letters

Is this onto? Because there is at least one word that starts with each letter, you can get all possible values in the Co-Domain, so it is onto.

Is this one-to-one? Well, no. For example, apple and alligator will give the same return value even though they are different words.