# Inverse Function Calculator

Inverse function calculator help you determine inverse values by providing inputs for the functions. Inverse functions are functions that can reverse other functions. Another name for it is anti-function. This may be expressed in the following way:

f(x) = y ⇔ f− 1(y) = x

## How to Use the Inverse Function Calculator?

It is extremely easy to use this online calculator to find inverse functions. For any function, you can find the inverse by following the steps below.

Step 1: Enter any function in the input box across the text “The inverse function of”.

Step 2: At the bottom of the calculator, click on the “Submit” button.

Step 3: A separate window will open in which you can compute the inverse of the given function.

## How to Find the Inverse of a Function?

The inverse of any function can be found by replacing the function variable with the other variable and then solving for the other variable by replacing it with the other variable. For a better understanding, here is an example.

Example: Inverse the expression f(x) = y = 3x − 2

Solution: Firstly, substitute f(x) with f(y).

Now, the equation y = 3x − 2 will become,

x = 3y − 2

Solve the problem for y,

y = (x + 2)/3

This gives y = (x + 2)/3 as the inverse of y = 3x − 2.

## Inverse function: What Is It?

Inverse functions, or anti-functions, are defined as functions that are able to reverse into another function. Basically, if any function takes an input x and transforms it into an output y, its inverse will do the same. In the case of a function denoted by “f” or “F”, the inverse function will be indicated by “f-1” or “F-1”. Please be careful not to confuse (-1) with exponents or reciprocals.

In the case of inverse functions, f(x) = y only if and only if g(y) = x

To find the angle for which a sine function produces a value, trigonometry uses the inverse sine function. In other words, sin-1(1) = sin-1(sin 90) = 90 degrees. Therefore, sin 90 degrees equals 1.

### Definition

An operation performs an operation on values, then creates an output based on these operations. Inverse functions agree with the resultant, operate and return the original function.

Inverse functions return the original value that is the output of a function.

If you consider f and g as functions, f(g(x)) = g(f(x)) = x. The original value of a function is obtained through its inverse.

Example: f(x) = 2x + 5 = y

Then, g(y) = (y-5)/2 = x is the inverse of f(x).

Note:

• Inverse relationship created by substituting the independent variable with a variable that is dependent on a specified equation, which may or may not be a function.
• f-1(x) denotes an inverse function if the inverse of a function is itself.

### Inverse Function Graph

Inverse graphs depict two things, one is the function and the other is the inverse of the function, over the line y = x. There is a slope value of 1 on this line, which passes through the origin. It can be expressed as;

y = f-1(x)

which is equal to; x = f(y)

The relation y = f(x) is somewhat similar to the graph of f, except the parts x and y are reversed. We must switch the positions of x and y on the axes if we are to draw the graph of f-1.

### How to Find the Inverse of a Function?

In general, inverses are calculated by swapping the coordinates x and y. Inverses are not necessarily functions, but they are relations.

To ensure that its inverse will also be a function, the original function must be a one-to-one function. A function is said to be one to one only if every second element corresponds to the first value (values of x and y are used only once).

Using the horizontal line test, you can determine whether a function is one-to-one. The function is a one-to-one function if a horizontal line intersects it in a single region and vice versa.

### Types of Inverse Function

Inverse functions include trigonometric functions, rational functions, hyperbolic functions, and logarithmic functions. Below are the inverses of some of the most common functions.

Inverse Trigonometric Functions

By obtaining the length of the arc required to obtain a particular value, the inverse trigonometric functions are also known as arc functions. These functions are arcsine (sin-1), arccosine (cos-1), arctangent (tan-1), arcsecant (sec-1), arccosecant (cosec-1), and arccotangent (cot-1).

Inverse Rational Function

f(x) is a rational function of the form P(x)/Q(x), where Q(x) ≠ 0. Inverse functions can be found by following the steps below. The following example can also assist you in understanding the concept better.

Step 1: Replace f(x) = y

Step 2: Interchange x and y

Step 3: Solve for y in terms of x

Step 4: Replace y with f-1(x) and the inverse of the function is obtained.

Inverse Hyperbolic Functions

They are the reverses of hyperbolic functions, just as inverse trigonometric functions are. Sinh-1, cosh-1, tanh-1, csch-1, coth-1, and sech-1 are the six major inverse hyperbolic functions. If you want to learn more about these functions in detail, refer to the inverse hyperbolic functions formula.

Inverse Logarithmic Functions and Inverse Exponential Function

Inverse functions of exponential functions are the natural log functions. In order to better understand inverse exponential functions and logarithmic functions, please review the following example. Get a better idea of how to answer similar questions, and therefore learn how to solve problems.

### Inverse Functions Example

Example 1: Find the inverse of the function f(x) = ln(x – 2)

Solution: First, replace f(x) with y

So, y = ln(x – 2)

Replace the equation in exponential way , x – 2 = ey

Now, solving for x,

x = 2 + ey

Now, replace x with y and thus, f-1(x) = y = 2 + ey

Example 2: Solve: f(x) = 2x + 3, at x = 4

Solution: We have,

f(4) = 2 × 4 + 3

f(4) = 11

Now, let’s apply for reverse on 11.

f-1(11) = (11 – 3) / 2

f-1(11) = 4

Magically we get 4 again.

Therefore, f-1(f(4)) = 4

So, when we apply function f and its reverse f-1 gives the original value back again, i.e, f-1(f(x)) = x.

Example 3: Find the inverse for the function f(x) = (3x+2)/(x-1)

Solution: First, replace f(x) with y and the function becomes,

y = (3x+2)/(x-1)

By replacing x with y we get,

x = (3y+2)/(y-1)

Now, solve y in terms of x :

x (y – 1) = 3y + 2

=> xy – x = 3y +2

=> xy – 3y = 2 + x

=> y (x – 3) = 2 + x

=> y = (2 + x) / (x – 3)

So, y = f-1(x) = (x+2)/(x-3)

## Conclusion

This invert function calculator will guide you through the process of inverting functions. The inverse function equation can be calculated manually, but it inherently increases the uncertainty, which is why this handy inverse function calculator provides you with 100% error-free results quickly.

What is the inverse function?

A function that returns the original value from which an output has been calculated is known as an inverse function. In the case of a function f(x) that gives output y, the inverse of y, f-1(y), will give value x.

How to find the inverse of a function?

Suppose, f(x) = 2x + 3 is a function.

Let f(x) = 2x+3 = y

y = 2x+3

x = (y-3)/2 = f-1(y)

This is the inverse of f(x).

Are inverse function and reciprocal of function, same?

Contrary to popular belief, inverse functions and reciprocals are not the same thing. F-1(x) is the inverse of the function since it returns the original value from which the output was calculated. Whereas reciprocal of function is given by 1/f(x) or f(x)-1

For example, f(x) = 2x = y

f-1(y) = y/2 = x, is the inverse of f(x).

But, 1/f(x) = 1/2x = f(x)-1 is the reciprocal of function f(x).

What is the inverse of 1/x?

Let f(x) = 1/x = y

Then inverse of f(x) will be f-1(y).

f-1(y) = 1/x

How to solve inverse trigonometry function?

If we have to find the inverse of trigonometry function sin x = ½, then the value of x is equal to the angle, the sine function of which angle is ½.

As we know, sin 30° = ½.

Therefore, sin x = ½

x = sin-1(½) = sin-1 (sin 30°) = 30°

What is the difference between reciprocal & inverse function?

A reciprocal function never returns the original values, while an inverse function always returns them. As reciprocal functions, 1 / f(x) is represented as f(x)-1. F-1(x) is the notation for inverse functions.

How inverse function used for temperature conversion?

Inverse functions used to convert Celsius (C) back to Fahrenheit (F) and vice versa:

To convert Fahrenheit (F) to Celsius (C): f (F) = 5/9 * (F – 32)

The inverse function for Celsius to Fahrenheit: f-1(C) = (C*9/5) + 32