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Inverse Function Formula
In relation to the original function f, the inverse function is denoted by f1, 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 f1 and occurs only when f is both a oneone function and an onto function. It must be remembered that f1 is not f’s inverse. By combining the functions f and f1, one may get the domain value of x.
The Inverse Function Formula is (f o f1) (x) = (f1 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 f1 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 = f1(x), and
Students have f1(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
Mathematics is one of the most challenging and highscoring subjects. Students who use the learning resources curated by the Extramarks website can improve their study skills and attain their objectives. These reference materials by Extramarks have been carefully compiled in order to aid students in learning and understanding the Inverse Function Formula. The language is easy to understand so that students may learn more and get the most out of their experience.
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FAQs (Frequently Asked Questions)
1. What is the Inverse Function Formula?
The Inverse Function Formula, indicated by the symbol f1, only occurs when the function is both a oneone and an onto function. Students must keep in mind that f1 is not f’s inverse. The domain value of x is determined by the combination of the function f and its reciprocal, f1.
The Inverse Function Formula is (f o f1) (x) = (f1 o f) (x) = x