# Inverse Function Formula

## Inverse Function Formula

In relation to the original function f, the inverse function is denoted by f-1, and the domain of the original function becomes the domain of the Inverse Function Formula, and the domain of the supplied function becomes the domain of the Inverse Function Formula. The graph of the Inverse Function Formula is created by swapping (x, y) with (y, x) with reference to the line y = x.

## What Is the Inverse Function Formula?

The Inverse Function Formula f is represented as f-1 and occurs only when f is both a one-one function and an onto function. It must be remembered that f-1 is not f’s inverse. By combining the functions f and f-1, one may get the domain value of x.

The Inverse Function Formula is (f o f-1) (x) = (f-1 o f) (x) = x

The Inverse Function Formula F is to be termed an inverse function, each element in the range y ∈ Y must be mapped from some element x ∈ X in the domain set, and this type of relationship is sometimes referred to as an injunction relationship. Also, the supplied function’s inverse f-1 has a domain y ∈ Y that is associated with a different element x ∈X in the codomain set, and this type of connection with reference to the given function ‘f’ is an onto function or a surjective function. As a result, the Inverse Function Formula, which is both an injunction and a surjective function, is referred to as a bijective function.

Consider a function f, the domain of which is the set X, and the codomain of which is the set Y. If another function g with domain Y and codomain X exists, the function f is invertible. The symbols for these two functions are f(x) = Y and g(y) = X. In this situation, if the function f(x) is inverse, then its inverse function g(x) is unique.

The two functions are said to be inverses of one another if their intersection results in the identity function f(g(x))=x. If applying a function to x as input yields n outputs of y, then applying another function g to y should provide the value of x. As a result, a function’s inverse reverses the function. The domain of the provided function is transformed into the range of the inverse function, and the range of the supplied function is transformed into the domain of the inverse function.

### Steps to Find the Inverse Function

The procedures below will assist students in quickly determining the inverse of a function. In this section, students study the function f(x) = ax + b and attempt to discover its inverse using the procedures below.

For the given function

f(x) = ax + b, replace f(x) = y

For getting y = ax + b.

Replace the x with the y and the y with the x in the function.

y = ax + b

To obtain x = ay + b.

Here, obtain y and address the formula x = ay + b.

And students obtain y = (x – b/a

Finally, replace y = f-1(x), and

Students have f-1(x) = (x – b)/a.

### Identifying Inverse Functions From a Graph

Students can determine whether two functions are inverses of one another if the graphs of the two functions are provided. The two functions are said to be inverses of one another if the graphs of both functions are symmetric with respect to the line y = x. This is because, if (x, y) lies on the original function, then (y, x) lies on the inverse function of the function.

### Solved Examples Using Inverse Function Formula

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