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Inverse Function Calculator

This tool helps you find the inverse of a given mathematical function f(x) with a step-by-step solution.

Enter a function y = f(x) = Enter any expression or function to find its inverse.

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This Inverse Function Calculator finds the inverse of a given function by reversing the relationship between its input and output. Enter a function f(x), and the calculator determines its inverse, f⁻¹(x), when an inverse exists. It provides the calculation steps to help you understand how the inverse is obtained. This makes it easier to solve inverse-function problems and verify your results.

What is an Inverse Function?

An inverse function is a function that reverses the operation of the original function, returning the output back to its original input.

In Mathematical Terms

Mathematically, the relationship between a function and its inverse is expressed as:

If → f(x) = y , Then ⇔ f-1(y) = x

Applying a function and then its inverse gets you back to your starting number: f-1(f(x)) = x

How to Use This Inverse Function Calculator?

Follow these three steps to use this inverse of a function calculator:

Enter the function f(x) for which you want to find the inverse.
Click “CALCULATE” to find inverse function.
Review the result and steps to see how the inverse function was obtained.

What are the Steps to Find the Inverse Function?

Let's learn how to find the inverse of a function, f⁻¹(x), using an example.

1. Replace f(x) with y:

First, rewrite the given function by replacing the value of y with function notation f(x).

For Example: Take f(x) = 2x + 3 and rewrite it as:

y = 2x + 3

2. Swap the positions of x and y:

Switch every “x” and “y” in the equation. Where you see y, write x, and where you see x, write y.

Swap the positions in:

y = 2x + 3
x = 2y + 3

3. Solve for y:

Rearrange the new equation to isolate (y) on one side and apply the inverse operation (such as addition, subtraction, multiplication, division, or factoring).

Subtract 3 from both sides:

x - 3 = 2y

Divide both sides by 2:

x - 3 / 2 = y

Flip it around so y is on the left:

y = x - 3 / 2

4. Write the inverse notation:

In the last step, substitute the final isolated expression “y” with f⁻¹(x).

Replace (y) to get your final answer:

f⁻¹(x) = x - 3 / 2

Along with manual calculation, you can save both time and effort by using our inverse function calculator. It takes only a few seconds to do all the above steps for you.

What are the Types of Inverse Functions?

Inverse functions are classified into various types, which are given below:

▸ Linear Inverse Functions

Linear functions have the form \(f(x)=ax+b\). Their inverse is also a linear function and can be found by swapping \(x\) and \(y\) and solving for \(y\).

▸ Quadratic Inverse Functions

It usually has the form \(f(x)=ax^2+bx+c\). Their inverse involves a square root and may require restricting the domain to make the inverse a function.

▸ Exponential and Logarithmic Inverse Functions

Exponential and logarithmic functions are inverses of each other. For example, \(f(x)=a^x\) has the inverse \(f^{-1}(x)=\log_a(x)\).

▸ Trigonometric Inverse Functions

Trigonometric functions such as sine, cosine, and tangent have inverse functions when their domains are restricted. Their inverses are written as \(\sin^{-1}(x)\), \(\cos^{-1}(x)\), and \(\tan^{-1}(x)\).

▸ Rational Inverse Functions

Rational functions are formed using ratios of polynomials, such as \(f(x)=\frac{ax+b}{cx+d}\). Their inverse can usually be found by swapping \(x\) and \(y\) and solving for \(y\).

Inverse Function Table:

Function f(x)	Inverse Function f⁻¹(x)
7x	x/7
3x + 7	(x − 7) / 3
4x + 2	(x − 2) / 4
2x − 5	(x + 5) / 2
5x − 3	(x + 3) / 5
x/2 + 3	2x − 6
x²	√x
√x	x²
1/x	1/x
eˣ	ln(x)
ln(x)	eˣ
sin(x)	sin⁻¹(x)
cos(x)	cos⁻¹(x)
tan(x)	tan⁻¹(x)
(x + 5)/2	2x − 5

Frequently Asked Questions:

Can this Functions Inverse Calculator handle multi-variable functions?

Yes, this inverse function finder can handle certain multi-variable functions, depending on whether the function has a valid inverse.

Can all functions be inverted?

No, not all functions can be inverted across their domain, only one-to-one functions can be inverted. Some functions have an inverse to the restricted portion of their domain.

Is f⁻¹(x) the same as 1 / f(x)?

No. f⁻¹(x) and 1 / f(x) are not the same terms.

f⁻¹(x) is the inverse function, which reverses the operation of the original function, while 1/f(x) is its reciprocal.

What is the rule for the domain and range of an inverse function?

The domain of the original function becomes the range of the inverse. The range of the original function becomes the domain of the inverse. You can learn more about this topic with our domain and range calculator.

References:

From the Source Wikipedia: Inverse function, Definitions, Inverses and composition, Notation, Examples, Formula for the inverse, Properties, Generalizations.

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