Question Video: Defining an Injective Function Mathematics

What is a one-to-one function?

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Video Transcript

What is a one-to-one function?

In this question, we’re asked to recall the definition of a one-to-one function. These are also sometimes called injective functions. And there’s many different ways of wording the definitions of these types of functions. One way is to say that they are functions where the elements in their domain correspond to, or map to, distinctive elements in their codomain.

However, it can also be useful to see this written out in function notation. We would say that the function 𝑓 is an injective or one-to-one function if the following condition holds. For any two elements of the domain of 𝑓, that’s π‘₯ one and π‘₯ two, we have if 𝑓 of π‘₯ one is equal to 𝑓 of π‘₯ two, then we need π‘₯ one to be equal to π‘₯ two. And this is exactly the same as the statement given. If 𝑓 evaluated at π‘₯ one is equal to 𝑓 evaluated at π‘₯ two, then we must have that π‘₯ one is equal to π‘₯ two. In other words, every element of the range of this function corresponds to exactly one element of the domain of the function.

Therefore, we were able to define a one-to-one function in this question. It is a function where the elements in its domain correspond to, or map to, distinctive elements in its codomain.

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