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 𝑥 one and 𝑥 two, we have if 𝑓 of 𝑥 one is equal to 𝑓 of 𝑥 two,
then we need 𝑥 one to be equal to 𝑥 two. And this is exactly the same as the
statement given. If 𝑓 evaluated at 𝑥 one is equal
to 𝑓 evaluated at 𝑥 two, then we must have that 𝑥 one is equal to 𝑥 two. In other words, every element of
the range of this function corresponds to exactly one element of the domain of the
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.