Inverse function. Theory and Application

In mathematics, inverse functions are mutually corresponding expressions that turn into each other. To understand what this means, it is worth considering a specific example. Suppose we have y = cos (x). If we take the cosine from the argument, then we can find the value of y. Obviously, for this you need to have X. But what if the game was originally given? This is where it comes to the heart of the matter. To solve the problem requires the use of the inverse function. In our case, it is the arccosine.

After all the transformations we get: x = arccos (y).

That is, to find the function opposite to the given one, it is enough to simply express an argument from it. But this only works if the result obtained has a unique meaning (more on this later).

In general terms, this fact can be written as follows: f (x) = y, g (y) = x.

Definition

Let f be a function whose domain of definition is the set X, and the domain of values ​​is the set Y. Then, if there exists g whose domains perform the opposite tasks, then f is invertible.

In addition, in this case, g is unique, which means that there is exactly one function satisfying this property (no more, no less). Then it is called the inverse function, and in a letter it is denoted as follows: g (x) = f ^-1 (x).

In other words, they can be considered as a binary relation. Invertibility takes place only when one value from another corresponds to one element of the set.

The inverse function does not always exist. To do this, each element y є Y must correspond to no more than one x є X. Then f is called one-to-one or injection. If f ^-1 belongs to Y, then each element of this set must correspond to some x ∈ X. Functions with this property are called surjections. It is carried out by definition if Y is an image f, but this is not always the case. To be inverse, the function must be both an injection and a surjection. Such expressions are called bijections.

Example: square and root functions

The function is defined on [0, ∞) and given by the formula f (x) = x ² .

Then it is not injective, since each possible result of Y (except 0) corresponds to two different X - one positive and one negative, therefore it is not reversible. In this case, it will be impossible to obtain the initial data from the received data, which contradicts the theory. She will be non-injective.

If the domain of definition by condition is limited to non-negative values, then everything will work as before. Then it is bijective and, therefore, reversible. The inverse function here is called positive.

Note on record

Let the notation f ^-1 (x) can confuse a person, but in no case can you use it like this: (f (x)) ^{- 1} . It refers to a completely different mathematical concept and has nothing to do with the inverse function.

In accordance with the general rules, some authors use expressions of the type sin ^-1 (x).

However, other mathematicians believe that this can cause confusion. To avoid such difficulties, inverse trigonometric functions are often denoted by the prefix "arc" (c Latin arc). In our case, we are talking about the arcsine. You can also occasionally see the prefix "ar" or "inv" for some other functions.