An Inverse function calculator computes the inverse value of any input function. The calculator also helps find the results of inverse functions. The user inputs their required function to the calculator. And this will fetch the results. In addition, an inverse function is a type of function which can invert another function. It is also called an anti-function. Moreover, it can reverse the operation of a specific function. For reference, an inverse function is denoted as follows:

**f(x) = y ⇔ ****(y) = x**

**How to Use the Inverse Function Calculator?**

The Inverse Function Calculator finds the inverse function of an input function. It is also an extremely easy tool. Moreover, this tool is available online for free use. While you can follow the given steps, the calculator will find the inverse of any function that you input.

- Step 1: Input any function in the input box. This input box is located across “The inverse function of” text on the screen.
- Step 2: Click on the “Submit” button which is present at the bottom of the calculator.
- Step 3: Then wait for the calculator to show a separate window. And this window, in turn, shall contain the answer to the input function. The final result will be the inverted form of the input function.

**Inverse Function Calculator eMath**

This calculator calculates the inverse of real functions. Moreover, a one-to-one function produces a unique inverse. This calculator is a very useful tool. It is also available to all users. Moreover, it is free for use. It is also very easy to use. You have to input the required function in the input box and the result will be shown.

Below is an image from eMath that explains what is a one-to-one function.

**Inverse Function Calculator mathway**

Mathway is a website designed by cheggs. Mathway also helps find the inverse of an input function. You can also use the mathway inverse function calculator to solve other problems too. While the primary use is to find inverses, other problems can also be solved.

**Inverse Function Calculator Symbolab**

Symbolab has an amazing calculator. Symbolab’s inverse function calculator is also extremely easy to use. Moreover, symbolab has an easy interface. The users can input their desired function. While the input is smooth, the results also appear nicely. Therefore, it is a well sought-after inverse function calculator to use.

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

Follow the following steps to find the inverse of a given function. Firstly, replace the variable given in the function. You should replace it with the other variable. Secondly, solve for the other variable by replacing them with each other. An example is given below for your benefit.

**Example: **Find the inverse of f(x) = y = 2x − 1

**Solution:**

First, replace f(x) with f(y).

Now, the equation y = 2x − 1 will become,

x = 2y − 1

Then solve for y,

y = (x + 1)/2

Thus, the inverse of y = 2x − 1 is y = (x + 1)/2

**Example: **Find the inverse of f(x) = y = (x – 2)/5

**Solution:**

First, replace f(x) with f(y).

Then, the equation y = (x – 2)/5 will become,

x = (y – 2)/5

Finally, solve for y,

y = 5x + 2

Thus, the inverse of y = (x – 2)/5 is y = 5x + 2

**What is an Inverse Function?**

In the mathematical sense, the inverse function of a given function *f* is a function that reverses the operation of the function. It is also called the inverse of function *f*. Therefore, it simply undoes what the original function intends to do. Though the inverse of *f* exists if and only if *f* is bijective. And if it exists, it is denoted by f^{-1}.

An inverse function is also called an anti-function. We also define it as a function, which can invert the operation of the original function. Or, in easier language, if any function “f” takes x to y then, the inverse of “f” will take y to x. Moreover, one should not confuse (^{-1}) with exponent or reciprocal here. Here, it is only representing the opposite anti-function of the given function.

If you consider the following functions g and f. Where g and f are also inverse to each other:

g(f(x)) = f(g(x)) = x.

A function that comprises its inverse also fetches the original value of itself.

Example: f(x) = 4x + 9 = y

Then, g(y) = (y – 9)/4 = x is the inverse of f(x).

**Note:**

- The relation, resulted when the independent variable is exchanged with the variable which is dependent on a specified equation. Moreover, this result and this inverse may or may not be a proper function.
- If the inverse of a function is the original, then it is known as a special inverse function, denoted by f
^{-1}(x).

**Inverse Function Graph**

It is easier to plot an inverse function on a graph and check its solution. Moreover, the graph of the inverse of a function simply shows two things. One of them is the function line while the second is the inverse of the function. Thus, the graphs are denoted over the line y = x. Nevertheless, this line in the graph passes through the origin. Also, the slope value of this line is 1. It can be represented as;

y = f^{-1}(x) which is equal to;

x = f(y)

This relation is somewhat equivalent to y = f(x). Which defines the graph of f but the part of x and y are exchanged on the graph. So if you have to plot the graph of f^{-1}, then we have to exchange the positions of x and y in their representative axes.

**How to Find the Inverse Function Graph?**

Usually, the method of finding an inverse is by exchanging the coordinates of x and y. This freshly created inverse is a relation. But this relation might not necessarily be a function. Math allows us to have a relationship that might not be a proper function.

The original function must be a one-to-one function. This is specifically to assure that its inverse will also be a function. Simply, so that the inverse can be a function that can exist. A function is identified as a one-to-one function only if every second element corresponds to the first value. In other words, values of x and y are used only once and are not repeated.

Read Also: Improper Integral Calculator

The graph line can be plotted with the basic lines interchanging between themselves to create the solution. The graph of the inverse of a function simply shows two things. One of them is the function line while the second is the inverse of the function. The graphs are denoted over the line y = x. This line in the graph passes through the origin. The slope value of this line is 1.

The inverse of *f* exists if and only if *f* is bijective. And if it exists, it is denoted by f^{-1}. So if you have to plot the graph of f^{-1}, then we have to exchange the positions of x and y in their representative axes.

You can use the horizontal line test to verify whether a function is a one-to-one function. If a horizontal line intersects the original function in a single region, the function is a one-to-one function. It also confirms that the inverse is also a function.

**Types of Inverse Functions**

There are many types of inverse functions.

FUNCTION |
INVERSE |
COMMENTS |

+ | – | It is a simple reversal of the initial operations. |

x | / | It is possible for all cases other than division by zero. Division by 0 will create an irrational result. |

1/x | 1/y | Neither x nor y should be equal to 0 for the functions to be real. |

x^{2} |
√y | x and y have to be positive. That means they have to be greater than or equal to 0. |

x^{n} |
y^{1/n} |
For the functions to remain rational, n cannot be zero. |

e^{x} |
Ln (y) | y should be greater than 0 for the function to exist |

a^{x} |
Log_{a }(y) |
Both y and a have to be greater than 0 to produce a rational function. |

Sin x | Sin^{-1} y |
– π/2 to + π/2 is the domain |

Cos x | Cos^{-1} y |
0 to π is the domain |

Tan x | Tan^{-1} y |
– π/2 to + π/2 is the domain |

**Inverse Trigonometric Functions**

The inverse trigonometric functions are also known as arc functions. Inverse Trigonometric Functions produce the length of the arc. This is the length of the arc that is needed to obtain that specific value of the inverse. There are six inverse trigonometric functions which include:

- arcsine (sin
^{-1}) - arccosine (cos
^{-1}) - arctangent (tan
^{-1}) - arcsecant (sec
^{-1}) - arccosecant (cosec
^{-1}) - arccotangent (cot
^{-1})

While the trigonometric functions calculate the basic values of the ratios given, the inverse ones trace the angle back for the particular function.

**Inverse Rational Function**

A rational function is a function of the form f(x) = P(x)/Q(x). In such an equation Q(x) ≠ 0. To find the inverse of a rational function, try to follow the following steps.

For example; follow the given below which can help you to understand the function better.

**Step 1:**Replace f(x) with y- Step 2: Interchange the variables x and y
- Step 3: Solve for y in the terms of x
**Step 4:**Replace y with f^{-1}(x) and hence, the inverse of the function is found.

**Inverse Hyperbolic Functions**

Similar to the inverse trigonometric functions, the inverse hyperbolic functions are the inverses of the simple hyperbolic functions. There are mainly 6 types of inverse hyperbolic functions;

- Sinh
^{-1} - Cosh
^{-1} - Tanh
^{-1} - Csch
^{-1} - Coth
^{-1} - Sech
^{-1}

There are easier formulae to understand the inverse hyperbolic functions.

Usually, the method of finding an inverse is by exchanging the coordinates of x and y. Therefore this freshly created inverse is called a relation. But this relation might not necessarily be a function. Nonetheless, math allows us to have a relation that might not be a proper function.

**Inverse Logarithmic Functions and Inverse Exponential Functions**

Natural logarithm functions serve as the inverses of specific exponential functions. To better comprehend inverse exponential functions and the logarithmic function, consider the following examples. These instances will offer more insight into solving similar problems and foster the development of your problem-solving skills.

Example 1: Exponential Function and its Inverse Consider the exponential function f(x) = e^x. The inverse function of f(x) is the natural logarithm function, denoted as g(x) = ln(x). Here, ‘e’ represents the mathematical constant (approximately 2.718). For these functions, if f(g(x)) = x, then g(f(x)) also equals x. In other words:

f(g(x)) = e^(ln(x)) = x g(f(x)) = ln(e^x) = x

Example 2: Logarithmic Function and its Properties The natural logarithm function, g(x) = ln(x), possesses several unique properties:

- ln(1) = 0, because e^0 = 1
- ln(e) = 1, because e^1 = e
- ln(ab) = ln(a) + ln(b), where a > 0 and b > 0
- ln(a/b) = ln(a) – ln(b), where a > 0 and b > 0
- ln(a^b) = b * ln(a), where a > 0

These properties help simplify expressions and solve equations involving natural logarithms.

By understanding the relationship between exponential and logarithmic functions, you can tackle various mathematical problems with ease. Developing a strong foundation in these concepts will enhance your problem-solving capabilities and make it easier to approach more complex questions.

**FAQs on Inverse Function Calculator**

**1. ****How do you find the inverse of a function on a calculator?**

To find the inverse of a function on an inverse function calculator, follow the given steps:

- Step 1: Input any function in the input box. This input box is located across “The inverse function of” text on the screen.
- Step 2: Click on the “Submit” button which is present at the bottom of the calculator.
- Step 3: Wait for the calculator to show a separate window. This window, in turn, shall contain the answer to the input function. Therefore, the final result will be the inverted form of the input function.

**2. ****How do you graph an inverse function step by step?**

On a graph, we will switch the axes and then plot the two lines. This relation is somewhat equivalent to y = f(x). Which defines the graph of f but the part of x and y are exchanged on the graph. So if you have to plot the graph of f^{-1}, then we have to exchange the positions of x and y and also their representative axes.

**3. ****What is the domain of the inverse function calculator?**

The domain of the inverse function depends on the function. The table earlier also shows the possible domains.

**4. ****What is the formula of an inverse function?**

There is no specific formula for an inverse function. Therefore, the solution is to be done manually or using the Inverse Function Calculator.

**5. ****What is the solution of an Inverse Function Calculator?**

The solution yields the result of the inverse function of the input function. This is also available on the online Inverse Function Calculator.

**6. ****What is the difference between a reciprocal and inverse function?**

A reciprocal function is simply 1/f where f is the function. While an inverse function undoes the operation of the function that was input.