F is a function if and only it f is one to one

correspondence between the x and y values : ) that is, for each and every x there is only one y values. It is a function if graphed and passes the straight line test. : )

What is your question?

To determine if a function f is one-to-one (also known as injective), we need to check if every element in the domain of f is mapped to a unique element in the codomain.

One way to verify if a function is one-to-one is to compare the outputs for different inputs. If each input produces a distinct output, then the function is one-to-one. However, this method may not be practical for all functions, especially when the domain and codomain are infinite sets.

Alternatively, we can use a proof by contradiction to show that a function is not one-to-one.

Assume that function f is not one-to-one. This means there exist two distinct elements x1 and x2 in the domain such that f(x1) = f(x2). This violates the definition of a function, where each input should have a unique output.

On the other hand, if we assume that f is one-to-one, we need to show that for any two distinct elements x1 and x2 in the domain, their corresponding outputs f(x1) and f(x2) are also distinct.

To do this, we can use a direct proof. Let's assume that f is one-to-one. If f(x1) = f(x2), where x1 and x2 are distinct elements, then by the definition of a one-to-one function, x1 = x2 must hold. However, since x1 and x2 are distinct, this leads to a contradiction.

Therefore, we can conclude that a function f is a one-to-one function (or injective) if and only if every input x1 and x2 in the domain has distinct outputs f(x1) and f(x2), or equivalently, f(x1) ≠ f(x2) whenever x1 ≠ x2.