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. Let's say 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 is initially given? This is where it gets to the heart of the matter. To solve the problem, the use of an inverse function is required. In our case, this is the arc cosine.
After all the transformations, we get: x=arccos(y).
That is, to find a function inverse to a given one, it is enough just to express an argument from it. But this only works if the result will have a single value (more on that 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 is the set X, andthe range of values is the set Y. Then, if there exists g whose domains perform opposite tasks, then f is reversible.
Besides, in this case g is unique, which means that there is exactly one function that satisfies this property (no more, no less). Then it is called the inverse function, and in writing it is denoted as follows: g(x)=f -1(x).
In other words, they can be viewed as a binary relation. Reversibility takes place only when one element of the set corresponds to one value from another.
There is not always an inverse function. To do this, each element y є Y must correspond to at most 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 holds by definition if Y is an image f, but this is not always the case. To be inverse, a 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)=x2.
Then it is not injective, because every possible outcome Y (except 0) corresponds to two different X's - one positive and one negative, so it is not reversible. In this case, it will be impossible to obtain the initial data from the received ones, which contradictstheories. It will be non-injective.
If the domain of definition is conditionally limited to non-negative values, then everything will work as before. Then it is bijective and hence invertible. The inverse function here is called positive.
Note on entry
Let the designation f -1 (x) may confuse a person, but in no case should it be used like this: (f (x))- 1 . It refers to a completely different mathematical concept and has nothing to do with the inverse function.
As a general rule, some authors use expressions like sin-1 (x).
However, other mathematicians believe that this can cause confusion. To avoid such difficulties, inverse trigonometric functions are often denoted with the prefix "arc" (from the 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.
