Inverse Function
Inverse function is represented by f^{1} with regards to the original function f and the domain of the original function becomes the range of inverse function and the range of the given function becomes the domain of the inverse function. The graph of the inverse function is obtained by swapping (x, y) with (y, x) with reference to the line y = x.
Let us learn more about the inverse function, the steps to find the inverse function, and the graph of inverse function.
1.  What Is Inverse Function? 
2.  Steps to Find An Inverse Function 
3.  Graph of An Inverse Function 
4.  Examples of Inverse Function 
5.  Practice Questions on Inverse Function 
6.  FAQs on Inverse Function 
What Is Inverse Function?
The inverse of a function f is denoted by f^{1} and it exists only when f is both oneone and onto function. Note that f^{1} is NOT the reciprocal of f. The composition of the function f and the reciprocal function f^{1} gives the domain value of x.
(f o f^{1}) (x) = (f^{1} o f) (x) = x
For a function 'f' to be considered an inverse function, each element in the range y ∈ Y has been mapped from some element x ∈ X in the domain set, and such a relation is called a oneone relation or an injunction relation. Also the inverse f^{1} of the given function has a domain y ∈ Y is related to a distinct element x ∈ X in the codomain set, and this kind of relationship with reference to the given function 'f' is an onto function or a surjection function. Thus the inverse function being an injunctive and a surjection function, is called a bijective function.
Let us consider a function f whose domain is the set X and the codomain is the set Y. The function f is invertible if there exists another function g whose domain is Y and its codomain is X. These two functions can be represented as f(x) = Y, and g(y) = X. For this situation, if the function f(x) is inverse, then its inverse function g(x) is unique.
If the composition of two functions f(x), and g(x), results in an identity function f(g(x))= x, then the two functions are said to be inverses of each other. If the application of a function to x as input gives n output of y, then the application of another function g to y should give back the value of x. Hence the inverse of a function reverses the function. The domain of the given function becomes the range of the inverse function, and the range of the given function becomes the domain of the inverse function.
Steps To Find An Inverse Function
The following sequence of steps would help in conveniently finding the inverse of a function. Here we consider a function f(x) = ax + b, and aim at finding the inverse of this function through the following steps.
 For the given function f(x) = ax + b, replace f(x) = y, to obtain y = ax + b.
 Interchange the x with y and the y with x in the function y = ax + b to obtain x = ay + b.
 Here solve the expression x = ay + b for y. And we obtain y = (x  b/a
 Finlly replace y = f^{1}(x), and we have f^{1}(x) = (x  b)/a.
Graph of An Inverse Function
The injective function is the reflection of the origin function with reference to the line y = x, and is obtained by swapping (x, y) with the (y, x).
If the graphs of two functions are given, we can identify whether they are inverses of each other. If the graphs of both functions are symmetric with respect to the line y = x, then we say that the two functions are inverses of each other. This is because of the fact that if (x, y) lies on the function, then (y, x) lies on its inverse function.
Related Topics
The following topics help in a better understanding of inverse function.
Examples of Inverse Function

Example 1: Find if the functions f(x) = 2x + 3 and g(x) = (x  3) /2 are inverses of each other using the inverse function formula.
Solution:
We will verify whether (f o g)(x) = (g o f)(x) = x.
(f o g)(x) = f (g(x))
= f ((x3)/2)
= 2 ((x3)/) + 3
= x  3 + 3
= x
(g o f)(x) = g (f(x))
= g (2x + 3)
= (2x + 3  3)/2
= 2x/2
= x
We proved that (f o g)(x) = (g o f)(x) = x.
By inverse function formula, f and g are inverses of each other.
Answer: We proved that f = g^{1} and g = f^{1}.

Example 2 : Find the inverse function for the given function f(x) = \( \dfrac{4x+1}{3x2}\).
Solution:
We will use the inverse function formula (or steps to find the inverse function).
The given function is:
y = \( \dfrac{4x+1}{3x  2}\)
Interchange x and y.
x = \( \dfrac{4y+1}{3y  2}\)
Now we will solve this for y.
Multiplying both sides by (3x  2),
x (3y  2) = 4y + 1
3xy  2x = 4y + 1
3xy  4y = 2x + 1
y (3x  4) = 2x + 1
y = \(\dfrac{2x + 1}{3x4}\)
Replace y with f^{1}(x).
f^{1}(x) = \(\dfrac{2x + 1}{3x4}\)
Answer: f^{1}(x) = \(\dfrac{2x + 1}{3x4}\).
FAQs on Inverse Function
What Is An Inverse Function?
The inverse of a function f is denoted by f^{1} and it exists only when f is both oneone and onto function. Note that f^{1} is NOT the reciprocal of f. The composition of the function f and the reciprocal function f^{1} gives the domain value of x.
(f o f^{1}) (x) = (f^{1} o f) (x) = x
How Do You Find the Inverse of a Function?
The following sequence of steps help in finding the inverse of a function. Here we consider a function f(x) = ax + b.
 For the given function f(x) = ax + b, replace f(x) = y, to obtain y = ax + b.
 Interchange the x with y and the y with x in the function y = ax + b to obtain x = ay + b.
 Here solve the expression x = ay + b for y. And we obtain y = (x  b/a
 Finlly replace y = f^{1}(x), and we have f^{1}(x) = (x  b)/a.
What Is an Example of An Inverse Function?
The example of a inverse function is a function f(x) = 2x + 3, and its inverse function is f^{1}(x) = (x  3)/2.
How To Find Inverse Function of Trigonometric Functions?
The inverse function of a trigonometric function is similar to finding the inverse of a normal function with algebraic expressions.
What Is the Domain and Range of An Inverse Function?
The domain and range of an inverse function is obtained by swapping the domain and range of the given function. The domain of the given function becomes the range of the inverse function, and the range of the given function becomes the domain of the inverse function.
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