What is a function?
In this question, we’re asked to recall the definition of a function. And in fact, there’s a lot of different ways of defining a function. We’ll go through a few of the common ones here. First, we can say that a function takes every member of a set and maps it onto exactly one member in a second set. And there’s a few important things we need to note about this definition. First, we need to know that the first set and the second set don’t need to be different sets. They can be the same set. For example, we can consider the function 𝑓 of 𝑥 is equal to two 𝑥, where we input a real number and it outputs another real number. And this agrees with our definition. We can take any member in the set of real numbers, input it into our function to double it, and we get exactly one member in the second set.
But this is only one way of defining a function. We can notice that this definition is very similar to that of a relation. And this means we’ll be able to define a function in terms of a relation. We recall a relation, or more specifically a binary relation, over two sets 𝑋 and 𝑌 is a set of ordered pairs lowercase 𝑥, lowercase 𝑦, where lowercase 𝑥 is an element of the set 𝑋 and lowercase 𝑦 is an element of the set 𝑌. We can then say that 𝑥 is related to 𝑦 if the ordered pair 𝑥, 𝑦 lies in the set. And we can also use this to define a function. The first value in our ordered pair will be the input value. And the second value in our set will be the output value.
However, we do need to be careful because not every relation will give a function. Because, remember, a function needs to take every member of the first set and map it onto exactly one member of the second set. In our relation, we called the first set 𝑋 and the second set 𝑌. Therefore, for a relation to be a function, it needs to have two properties. Every element in 𝑋 needs to be related to an element in 𝑌, and it needs to be related to exactly one element of 𝑌. We can write this more succinctly as one property. A function is a relation which relates each input to exactly one output, which then allows us to answer the question “What is a function?” It’s a relation which relates each input to exactly one output.