Math Calculators ▶ Inverse Function Calculator
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Use this free online inverse function calculator with steps that helps you to determine the inverse of any given function, with a step-by-step solution. However, the inverse of a particular function may or may not be a function. Here you can see how to find the inverse of a function, inverse graph, and much more.
Let’s dive in!
In mathematics, an inverse function is a function (f) that inverts the particular function. The inverse function of (f) is represented as f-1.
f (y) = x ⇔ f−1(x) = y
The inverse function calculator with steps determines the inverse function, replaces the function with another variable, and then finds another variable through mutual exchange.
However, an Online Composite Function Calculator allows you to solve the composition of the functions from entered values of functions.
“A one to one function is the one whose each element in range maps an element in domain.”
A function will have an inverse if it is one to one function. For further clarification of concept, let’s have a look at the following pictorial representation:
Now if you see the above picture, you will notice two functions are there. In the first function f(x), we have some numbers that are assigned to different variables. While in the second function, the variables are being mapped against the numbers that denote the inverses of the function f(x). This shows every function is structured back in its inverse by reversing each element in the layout. You can also analyse this behaviour of the function with the assistance of this simple one to one function calculator.
The graph of a function where f is invertible:
$$x = f^{-1} (y)$$
Which, is the same as the graph of an equation:
$$y = f (x)$$
The equation x= f(y) defines the graph of f, except that the roles of y and x have been reversed. Thus the graph for inverse function (f^{-1}) can be obtained from the graph of the function (f) by switching the position of the y and x-axis.
Below we have arranged a table display that highlights some most important functions along with their inverses. Let’s have a look!
Function f(x) |
Inverse f −1(y) | Notes |
x + a | y − a | |
a − x |
a − y | |
mx |
y/m | m ≠ 0 |
1/x (i.e. x−1) | 1/y (i.e. y−1) |
x, y ≠ 0 |
x2 |
{\displaystyle {\sqrt {y}}}
(i.e. y1/2) |
x, y ≥ 0 only |
x3 |
{\displaystyle {\sqrt[{3}]{y}}}
(i.e. y1/3) |
no restriction on x and y |
xp | {\displaystyle {\sqrt[{p}]{y}}}
(i.e. y1/p) |
x, y ≥ 0 if p is even; integer p > 0 |
2x |
lb y | y > 0 |
ex | ln y |
y > 0 |
10x |
logy | y > 0 |
ax | loga y |
y > 0 and a > 0 |
xex |
W (y) |
x ≥ −1 and y ≥ −1/e |
For further assistance, you can also use our free inverse of rational functions calculator.
Let’s go through the following characteristics of the inverse functions:
For any function, you can check behaviour of all these properties by using this swift how to find the inverse of a function calculator.
Compute the inverse function (f^{-1}) of the given function by the following steps:
To make it convenient for you, the inverse function calculator with steps does all these calculations for you in a fraction of a second.
Example # 01:
Calculate the inverse of the functions x = y+11/13y+19?
Solution:
Replace the variables y & x, to find inverse function f^{-1 }with inverse calculator with steps:
$$y = x + 11 / 13x + 19$$
$$y (13x + 19) = x + 11$$
$$13xy + 19y – x = 11$$
$$x (13y – 1) = 11 – 19y$$
$$x = 11 – 19y / 13y – 1$$
Hence, the inverse function of y+11/13y+19 is 11 – 19y / 13y – 1.
Here you can also verify the results by using this best find f^-1(x) calculator.
However, an Online Quadratic Formula Calculator helps to solve a given quadratic equation by using the quadratic equation formula.
Example # 02:
Determine the multi function inverse of the function if exists:
$$ f(x)=2x^3+1\text{.} $$
Solution:
As the given function is as follows:
$$ f(x)=2x^3+1\text{.} $$
Here we have:
$$ y-1 = 2x^{3} $$
$$ \frac{y-1}{2} = x^{3} $$
$$ x = \sqrt[3]{\frac{y-1}{2}} $$
Which is the required inverse of the function and can also be determined by using the find the inverse of the function calculator.
An online inverse of a function calculator finds the inverse of entered function with these steps:
Input:
Output:
Reciprocal functions are one which never returns the original values but the inverse functions always return the original values. Reciprocal functions are represented as f(x)^{-1} or 1 / f(x). Whereas inverse functions are denoted by f^{-1}(x) and can also be determined by the use of the inverse calculator. .
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
Suppose that:
$$f (y) = 1/y = x$$
Replace the y and x variables:
$$y = 1/x$$
$$f^{-1}(y) = 1/x$$
For further verification, you can make use of this best function inverse calculator.
An online find inverse function calculator provides a step-by-step solution for invert functions according to given values. Although you can calculate inverse manually with an inverse function equation, it increases the ambiguity so, this handy finding inverse functions calculator gives 100% error-free results quickly.
From the source of Wikipedia: Inverses and composition, Notation, Self-inverses, Graph of the inverse, Inverses, and derivatives.
From the source of Paul Online Notes: Inverse Functions, Finding the Inverse of Function calculator, Partial inverses, Left and right inverses.
From the source of Quest Calculus: DERIVATIVES OF INVERSE FUNCTIONS, The chain rule, Two-sided inverses, Preimages.