24.1.5. Inverse of a Function

Now we will define what we call as the inverse of a function. So, imagine a function f: A → B. Since f is a special type of relation, remember it is a special type of relation we can find the inverse of that relation as well.

In this chunk, we introduce the concept of the inverse of a function. A function is often viewed as a relation between two sets, say A and B. The inverse function, denoted f⁻¹, is a way of 'reversing' this mapping, meaning that if f maps an element 'a' in set A to an element 'b' in set B, then the inverse function will map 'b' back to 'a'. It's crucial to understand that this inverse function must still satisfy the definition of a function, meaning it must provide a unique output for each input.

Think of a function like a vending machine: you select a number (input), and it dispenses an item (output). The inverse function would be like having a way to trace back from the item you received to the number you pressed. If you wanted a specific snack and you only have that, you cannot select a different number without knowing what you pressed originally. Thus, the inverse function serves a similar purpose of tracing back the mapping.

It is easy to see that a function f will be invertible if and only if the function is a bijection. If the function f is not a bijection then, we cannot define the inverse of that function.

In this chunk, we discuss the conditions under which a function can be inverted. A function is said to be a bijection if it is both injective (one-to-one) and surjective (onto). This means that every element from set A maps to a unique element in set B (injective) and every element in B has at least one corresponding element in A (surjective). If a function does not meet these criteria, it cannot have an inverse function.

Consider the relationship between students in a class and their student IDs. If each student has a unique ID (injective), and every ID corresponds to a student (surjective), we can easily find a student based on their ID and vice versa. However, if some IDs are shared between students, then identifying the student based solely on an ID becomes impossible. Hence, the association loses its invertibility.

So, let us prove this formally. We can show that if your function f is not an injective function then clearly f is not invertible.

This section formalizes the proof that if a function is not injective, then it is not invertible. If a function maps two different elements in A to the same element in B, when we attempt to find the inverse, we would find that both elements map to the same output. Therefore, we couldn't determine the unique original input from this output, leading to a contradiction in the definition of a function.

Imagine a library system where two books share the same title ('The Great Gatsby'), but are different editions. If someone asks for the book with that title and the ID system only records the title without clear distinction of edition, it becomes unclear which copy of 'The Great Gatsby' is being referred to. This ambiguity illustrates that the inverse function cannot be clearly defined without unique identifiers.

We can also prove here that if the function f is not a surjective function then also it is not invertible.

In this chunk, we discuss that if a function is not surjective, there exists at least one element in set B that is not mapped from any element in set A. As a result, when attempting to reverse this mapping, there would be elements in B that do not correspond to any input in A, leading to undefined outputs in the inverse function.

Picture a restaurant menu where certain dishes are not available (not surjective). If you are told that a dish was served (say, 'Lasagna'), but you never ordered it because it wasn't on the menu, you cannot trace back to what you ordered. Therefore, if a dish isn't available, it cannot have an inverse operation—leading again to a situation where defining the inverse is simply not possible.

So, that shows that a function is invertible if and only if f is a bijection.

This is a concluding statement affirming that a function can only be inverted if it meets the criteria of being a bijection, which encompasses both injectiveness and surjectiveness. This understanding is pivotal in identifying functions that possess a clear dual relationship between inputs and outputs, thereby enabling the establishment of their respective inverses.

Returning to our vending machine analogy, this time imagine a model that guarantees every item dispensed has its unique selection button and every button results in an item that exists. This set-up assures that each action in the vending machine can reverse, giving clarity and precision to the vending experience.

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