Invertible Functions

Functions that can inverse other functions are known as inverse functions. In mathematical terms, an invertible function for the given function f is one that can undo the operation. However, for the invertible function to exist, f needs to be bijective, expressed as f-1. 

We can discover a function g: B-> A such that g(y) = x when y = f if the function f: A-> B is both one to one and onto, i.e., bijective (x). The inverse of f is called g, symbolised as f-1.  Here, function f(x) is the invertible function.

Definition of Invertible Function

The Function f: A-> B can only be invertible if we can find a function g: B- > A such that fog = y and  gof = x

F-1 denotes the inverse of a function f, which occurs only when f is both one-one and onto. It is vital to remember that f-1 is not the same as f. Composing the function f with the reciprocal function f-1 yields the domain value of x.

(fo f-1) (x) = (f-1 o f) (x) = x

To be termed an inverse function, each element in the range yY has been mapped from some element xX in the domain set, and this relation is known as a one-one relation or an injunction relation. 

Furthermore, the inverse f-1 of the supplied function has a domain yY associated with a distinct element xX in the codomain set. This relationship concerning the given function ‘f’ is known as an onto function or a surjective function. As a result, the inverse function, both injection and surjection, is a bijective function.

How to Find the Inverse of a Function

The procedures below will assist you in quickly finding the inverse of a matrix. In this example, we will look at the function f(x) = ax + b and try to discover its inverse using the techniques below.

• The function of the given equation 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)
• Finally, replace y = f-1(x), and we have f-1(x) = (x – b)/a.

We can find the inverse of a function using one of three ways:

1. Simply swapping the ordered pairs
2. Solve it algebraically
3. Using a graph

Finding the Inverse Graph: An inverse function’s graph reflects the original graph across the identity line y = x.

Types of Inverse Functions

Inverse functions include the inverse of rational functions, log functions, trigonometric functions,  and hyperbolic functions, among others.

Inverse Trigonometric Functions

Inverse trigonometric functions are sometimes known as arc functions since they yield the arc length needed to acquire a given number. Arccosine (cos-1),  arccosecant (cosec-1), arccotangent  (cot-1), arctangent (tan-1), arcsecant (sec-1), and arcsine (sin-1), are the six inverse trigonometric functions.

Inverse Rational Function

A rational function has the form f(x) = P(x)/Q(x), where Q(x) is less than 0. Follow the instructions below to obtain the inverse of a rational function. Below is an example that will assist you in better understanding the topic.

Step 1: Change f(x) to y.

Step 2: Swap the x and y coordinates.

Step 3: Find y in terms of x.

Step 4: Substitute f-1(x) for y to get the inverse of the function.

Inverse Hyperbolic Functions

Inverse hyperbolic functions are the inverses of hyperbolic functions, just as inverse trigonometric functions. Sinh-1, cosh-1, tanh-1, csch-1, coth-1, and sech-1 are the six most common inverse hyperbolic functions. To understand more about these functions, look into the inverse hyperbolic functions formula.

Inverse Logarithmic Functions and Inverse Exponential Function

The exponential functions are inverses by the natural log functions. Look at the examples below to learn more about the inverse exponential and logarithmic functions. Also, learn more about overcoming similar problems and improving your problem-solving abilities.

An inverse to a function does not always exist. Each element b∈B must not have more than one 

a∈ A for a function to have an inverse. The function must be of the injective type. 

In addition, we should be able to map every element of B to an element of A. The function must be subjective. The function must be one-one and onto, and vice versa, to be invertible.

Determining if a Function is Invertible

In general, the inverse is calculated by switching the x and y coordinates. This new inverse equation  is a relation rather than a function.

To ensure that the inverse is likewise a function, the original function must be a one-to-one function. Only if every second element matches to the first value is a function considered to be one to one (values of x and y are used only once).  So let us look at some of the difficulties to see how we might decide whether a function is invertible or not.

Invertible Functions: 

A function f from a set X to a set Y is said to be invertible if for every y in Y and x in X, there exists a function g from Y to X such that f(g(y)) = y and g(f(x)) = x. 

In other words, the invertible function is a function from B to A that has the feature that can go  from A to B and B to A where every element of the first set returns to itself.

Invertible functions have an inverse; thus, they are uniquely specified and denoted by -1. 

Take, for instance,

Example 1: Assume A = 1, 2, 3 and B = a, b, c, d. 

Take a look at the function f = (1, a), (2, b), (3, c). 

The image set or range, in this case, is a, b, c, or d, which is not the same as the codomains a, b, c, or d. 

Conclusion

Here we learned about the invertible functions with an elaborated explanation. We covered the step-by-step method used in finding the inverse of a function as well. The examples in the article assist in better understanding the calculation and the concept.