Question

For each of the functions below, indicate whether the function is onto, one-to-one, neither or both. If the function is not onto or not one-to-one, give an example showing why?

1) Onto
2) One-to-one
3) Neither
4) Both

Answer 1

For each of the functions, the type of function (onto, one-to-one, neither, or both) is determined. Examples are provided to illustrate each case.

Step-by-step explanation:

1) Onto: A function is onto if every element in the codomain has a pre-image in the domain. To determine if a function is onto, we need to check if every element in the codomain is being mapped by the function. If there is at least one element in the codomain that has no pre-image in the domain, the function is not onto. Example: Let's say the codomain is the set of all real numbers and the function maps every real number to its square. Since every real number has a square root, this function is onto.

2) One-to-one: A function is one-to-one (or injective) if no two different elements in the domain are mapped to the same element in the codomain. To determine if a function is one-to-one, we need to check if there are no two distinct elements in the domain that map to the same element in the codomain. If there are at least two distinct elements in the domain that map to the same element in the codomain, the function is not one-to-one. Example: Let's say the domain is the set of all real numbers and the function maps every real number to its square. Since different real numbers can have the same square, this function is not one-to-one.

3) Neither: If a function is neither onto nor one-to-one, it means there are elements in the codomain that have no pre-image in the domain and there are also two distinct elements in the domain that map to the same element in the codomain. Example: Let's say the domain is the set of all real numbers and the function maps every real number to its absolute value. This function is neither onto nor one-to-one because the absolute value of a negative real number is the same as the absolute value of its positive counterpart, causing different elements in the domain to map to the same element in the codomain.

4) Both: For a function to be both onto and one-to-one, it needs to be a bijection, which means every element in the codomain has a unique pre-image in the domain. Example: Let's say the domain is the set of all real numbers and the function maps every real number to its cube. Since every real number has a unique cube root, this function is both onto and one-to-one.