The Conceptual Theory of Inverse Function

An inverse function is a mathematical function, which can reverse into another function. Learn how to find the inverse of a function.

Understanding the Concept of Inverse Functions

In order to get the inverse of any given function, inverse operations are always used in mathematical analysis. A logarithmic function, for example, is one that may be used to calculate the inverse of an exponential function. In any case, our major goal is to demonstrate how to exactly locate the inverse of any expression. However, if you need accurate and immediate results, you try an online inverse function calculator. You can find the inverse of any mathematical function in a blink of an eye by utilizing any free inverse calculator. So what are your thoughts about this?

The procedure for determining the inverse of a function will be discussed in this technical read below.

Let's get going!

Inverse Function
Theory of Inverse Function

What Is a Function's Inverse?

In mathematics, an inverse is a function that aids in the "undoing" of another function. If f(x) creates y, then inserting y into the inverse of f yields x in this case. An invertible function f has an inverse, which is denoted by the symbol f-1.

The Inverse of a Function Equation: 

The following formula may be used to get the inverse of a function:

f (y) = x ⇔ f−1(x) = y

However, instead of using this method, which takes a long time to calculate the answer, we recommend using the online inverse calculator for quick and accurate answers.

Steps Involved:

Using an inverse calculator is the most efficient approach to get the inverse of a function. Manual calculations, on the other hand, are a little more difficult. But don't worry, we'll explain everything and make it extremely simple for you. The three key approaches that you must deal with in order to handle inverse function problems are described below.

  • To begin, replace f(x) with y, then all x and y with y, then all y with the x.
  • After that, calculate the expression for the variable y that results. Just be cautious, as this phase is quite scary and may result in errors during calculations.

Some Standard Inverses:

Below we have summarized the most useful functions and their inverses that can also be verified by using the free inverse calculator in a fraction of seconds. 

Function f(x)

Inverse f −1(y)


x + a

y a

a − x

a − y



m ≠ 0

1/x(i.e. x−1)

1/y (i.e. y−1)

x, y ≠ 0


lb y

y > 0


ln y

y > 0


log y

y > 0


loga y

y > 0 and a > 0



x ≥ −1 and y ≥ −1/e

The Inverse Function's Graph:

One thing to keep in mind is that the graphs of the functions x = f-1(y) and y = f(x) are identical. What has just happened is that the variables x and y have been changed in order to generate a graph of the formula y = f (x).

Inverse Function Graph
Inverse Function Graph

What Is The Inverse Of Any Function? (Instructions)

This part, on the other hand, is in search of a focus. We'll resolve a few cases right now to help you better understand your concept. Let's look into it.

Example # 01:

Find the function's inverse here:

y = x + 4x


First and foremost in the provided function, we will flip the variables x and y as follows:

y = x + 4x

x = y + 4y

x - y = 4y

-y = 4y -x

y = x - 4y

Which is the needed inverse of the provided function, which can also be determined in seconds with the help of the free online inverse function calculator.

Example # 02:

Find the inverse of the following function:

x = y+11/13y+19


To create the results, we'll repeat all of the processes we used in example # 01:

x = y+11/13y+19

y = x + 11/13x + 19

y(13x + 19) = x + 11

13xy + 19y = x + 11

13xy + 19y - x = 11

13xy - x = 11 - 19y

x(13y - 1) = 11 - 19y

x = 11 - 19y/13y - 1

Calculate the inverse of a function using the Inverse Function Calculator:

To get better results for your issues, try the free inverse of a function calculator. Let's have a look at how we can do it!

  • In the designated field, list down your function.
  • Here's what you get if you press the calculate button.
  • The solution will appear on the calculator's screen right away.

It's incredible how quickly things happen.

Real-Life Examples:

Distance is a function of time and speed, as you know. You use the inverse of the aforementioned function when you know the distance and speed and want to know how long it will take you to reach your destination. The division is the inverse of multiplication, to put it another way. Inverse functions are used often in our everyday lives. We just aren't aware of it because we've previously defined an inverse as a common function. But we are sure that you can understand the real-world problems of the inverse function by using the free inverse calculator. 

Inverse functions are employed in real life on a daily basis. When a computer receives a number you punch in, it transforms it to binary for internal storage before printing it out again on the screen you see - this is an example of an inverse function. Converting temperature from ℉ to is a simple example.

Another example: 

If music notes on paper are a function of the sound created, then the program Sibelius may be thought of as the inverse function, as it turns a musician's music back to music notes. 


But wait wait, besides all these examples, the best and most reliable one is the proposal of the free inverse calculator. Its parent website,, is continuously striving for providing the best solution to complicated mathematical problems within a few clicks. This is indeed one of the greatest opportunities available for professionals and students. 

Read Here: How To Calculate Percentage: Introduction, Formula & Uses

To Sum Up: 

In this article, we looked at how to get the inverse of a function. It's also been mentioned how to speed up these computations by utilizing a free inverse function calculator. We hope it will be of great assistance to you!

Read Also: How to Convert Different Types of Numbers in Standard Form

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The Scientific World is a Scientific and Technical Information Network that provides readers with informative & educational blogs and articles. Site Admin: Mahtab Alam Quddusi - Blogger, writer and digital publisher.

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