Both is correct.ver_mathstats said:
Thank you.fresh_42 said:
A useful denial of an injective function is when the function is not one-to-one, meaning that there are two or more inputs that produce the same output. This can be useful for creating a more efficient or simplified function.
To prove that a function is not injective, you can provide a counterexample where two different inputs produce the same output. You can also use the horizontal line test, where a horizontal line intersects the graph of the function more than once, indicating that the function is not one-to-one.
A useful denial of a surjective function is when the function is not onto, meaning that there are elements in the range that are not mapped to by any element in the domain. This can be useful for creating a function with a smaller range or for simplifying the function.
To prove that a function is not surjective, you can provide a counterexample where an element in the range is not mapped to by any element in the domain. You can also use the vertical line test, where a vertical line intersects the graph of the function more than once, indicating that the function is not onto.
Yes, a function can be both injective and surjective. This type of function is called a bijection and it means that every element in the domain is mapped to a unique element in the range, and every element in the range is mapped to by at least one element in the domain.
