## Inverse Functions

An **inverse function** or an anti function is defined as a function, which can reverse into another function. In simple words, if any function “f” takes x to y then, the inverse of “f” will take y to x. If the function is denoted by ‘f’ or ‘F’, then the inverse function is denoted by f^{-1} or F^{-1}. One should not confuse (-1) with exponent or reciprocal here.

If f and g are inverse functions, then f(x) = y if and only if g(y) = x |

In trigonometry, the inverse sine function is used to find the measure of angle for which sine function generated the value. For example, sin^{-1}(1) = sin^{-1}(sin 90) = 90 degrees. Hence, sin 90 degrees is equal to 1.

## Definition

A function accepts values, performs particular operations on these values and generates an output. The inverse function agrees with the resultant, operates and reaches back to the original function.

The inverse function returns the original value for which a function gave the output. |

If you consider functions, f and g are inverse, f(g(x)) = g(f(x)) = x. A function that consists of its inverse fetches the original value.

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

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

**Note:**

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

## Inverse Function Graph

The graph of the inverse of a function reflects two things, one is the function and second is the inverse of the function, over the line y = x. This line in the graph passes through the origin and has slope value 1. It can be represented as;

y = f^{-1}(x)

which is equal to;

x = f(y)

This relation is somewhat similar to y = f(x), which defines the graph of f but the part of x and y are reversed here. So if we have to draw the graph of f^{-1}, then we have to switch the positions of x and y in axes.

## How to Find the Inverse of a Function?

Generally, the method of calculating an inverse is swapping of coordinates x and y. This newly created inverse is a relation but not necessarily a function.

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

You can apply on 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 and inverse is also a function.

## Types of Inverse Function

There are various types of inverse functions like the inverse of trigonometric functions, rational functions, hyperbolic functions and log functions. The inverses of some of the most common functions are given below.

Function | Inverse of the Function | Comment |
---|---|---|

+ | – | |

× | / | Don’t divide by 0 |

1/x | 1/y | x and y not equal to 0 |

x^{2} | √y | x and y ≥ 0 |

x^{n} | y^{1/n} | n is not equal to 0 |

e^{x} | ln(y) | y > 0 |

a^{x} | log a(y) | y and a > 0 |

Sin (x) | Sin^{-1} (y) | – π/2 to + π/2 |

Cos (x) | Cos^{-1} (y) | 0 to π |

Tan (x) | Tan^{-1} (y) | – π/2 to + π/2 |

**Inverse Trigonometric Functions**

The inverse trigonometric functions are also known as** arc function** as they produce the length of the arc, which is required to obtain that particular value. There are six inverse trigonometric functions which include arcsine (sin^{-1}), arccosine (cos^{-1}), arctangent (tan^{-1}), arcsecant (sec^{-1}), arccosecant (cosec^{-1}), and arccotangent (cot^{-1}).

**Inverse Rational Function**

A rational function is a function of form f(x) = P(x)/Q(x) where Q(x) ≠ 0. To find the inverse of a rational function, follow the following steps. An example is also given below which can help you to understand 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**

Just like inverse trigonometric functions, the inverse hyperbolic functions are the inverses of the hyperbolic functions. There are mainly 6 inverse hyperbolic functions exist which include sinh^{-1}, cosh^{-1}, tanh^{-1}, csch^{-1}, coth^{-1}, and sech^{-1}. Check out inverse hyperbolic functions formula to learn more about these functions in detail.

**Inverse Logarithmic Functions and Inverse Exponential Function**

The natural log functions are inverse of the exponential functions. Check the following example to understand the inverse exponential function and logarithmic function in detail. Also, get more insights of how to solve similar questions and thus, develop problem-solving skills.

### Finding Inverse Function Using Algebra

Put “y” for “f(x)” and solve for x:

The function: | f(x) | = | 2x+3 |

Put “y” for “f(x)”: | y | = | 2x+3 |

Subtract 3 from both sides: | y-3 | = | 2x |

Divide both sides by 2: | (y-3)/2 | = | x |

Swap sides: | x | = | (y-3)/2 |

Solution (put “f^{-1}(y)” for “x”) : | f^{-1}(y) | = | (y-3)/2 |

### 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 = e^{y}

Now, solving for x,

x = 2 + e^{y}

Now, replace x with y and thus, f^{-1}(x) = y = 2 + e^{y}

**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)

## Frequently Asked Questions – FAQs

### What is the inverse function?

An inverse function is a function that returns the original value for which a function has given the output. If f(x) is a function which gives output y, then the inverse function of y, i.e. f^{-1}(y) will return the 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?

One should not get confused inverse function with reciprocal of function. The inverse of the function returns the original value, which was used to produce the output and is denoted by f^{-1}(x). 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°

## Inverse Functions

*An inverse function goes the other way!*

Let us start with an example:

Here we have the function **f(x) = 2x+3**, written as a flow diagram:

## Back to Where We Started

The cool thing about the inverse is that it should give us back the original value:

## Solve Using Algebra

We can work out the inverse using Algebra. **Put “y” for “f(x)” and solve for x:**

## Inverses of Common Functions

It has been easy so far, because we know the inverse of Multiply is Divide, and the inverse of Add is Subtract, but what about other functions?

Here is a list to help you:

So the square function (as it stands) **does not have an inverse**

### But we can fix that!

**Restrict the Domain** (the values that can go into a function).

### Example: (continued)

Just make sure we don’t use negative numbers.

In other words, restrict it to **x ≥ 0** and then we can have an inverse.

So we have this situation:

## Domain and Range

So what is all this talk about “**Restricting the Domain**“?

## Inverse Function Formula

Before learning the inverse function formula, let us recall what is an inverse function. If the composition of two functions results in an identity function (I(x) = x), then the two functions are said to be inverses of each other. The inverse of a function f is denoted by f^{-1} and it exists only when f is both one-one and onto function. Note that f^{-1} is NOT the reciprocal of f. Let us learn the inverse function formula in the upcoming sections.

## What Is the Inverse Function Formula?

If (x, y) is a point on the graph of a function f, then (y, x) will be definitely a point on f^{-1}, i.e., the domain of f is the range of f^{-1} and the range of f is the domain of f^{-1},i.e., if f : A → B (which is one-one and onto), then f^{-1}: B → A. The inverse function formula says f and f^{-1 }are inverses of each other only if their composition is x.

(f o f^{-1}) (x) = (f^{-1} o f) (x) = x

### Steps to Find the Inverse Function

Here are the steps to find the inverse of a function y = f(x).

- Interchange x and y.
- Solve for y.
- Replace y with f
^{-1}(x).

### Identifying Inverse Functions From a Graph

If the graphs of two functions are given, we can identify whether they are inverses of each other. If the graphs of both functions are symmetric with respect to the line y = x, then we say that the two functions are inverses of each other. This is because of the fact that if (x, y) lies on the function, then (y, x) lies on its inverse function.

Let us see some applications of the inverse function formula in the solved examples section.

## Solved Examples Using Inverse Function Formula

Example 1: Verify whether f(x) = 3x + 2 and g(x) = (x – 2) /3 are inverses of each other using the inverse function formula.

**Solution:**

We will verify whether (f o g)(x) = (g o f)(x) = x.

(f o g)(x) = f (g(x))

= f ((x-2)/3)

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

= x – 2 + 2

= x

(g o f)(x) = g (f(x))

= g (3x + 2)

= (3x + 2 – 2)/3

= 3x/3

= x

We proved that (f o g)(x) = (g o f)(x) = x.

By inverse function formula, f and g are inverses of each other.

**Answer:** **We proved that f = g ^{-1} and g = f^{-1}.**

## Inverse Function

Inverse function is represented by f^{-1} with regards to the original function f and the domain of the original function becomes the range of inverse function and the range of the given function becomes the domain of the inverse function. The graph of the inverse function is obtained by swapping (x, y) with (y, x) with reference to the line y = x.

Let us learn more about the inverse function, the steps to find the inverse function, and the graph of inverse function.

## What Is Inverse Function?

The inverse of a function f is denoted by f^{-1} and it exists only when f is both one-one and onto function. Note that f^{-1} is NOT the reciprocal of f. The composition of the function f and the reciprocal function f^{-1} gives the domain value of x.

(f o f^{-1}) (x) = (f^{-1} o f) (x) = x

For a function ‘f’ to be considered an inverse function, each element in the range y ∈ Y has been mapped from some element x ∈ X in the domain set, and such a relation is called a one-one relation or an injunction relation. Also the inverse f^{-1} of the given function has a domain y ∈ Y is related to a distinct element x ∈ X in the codomain set, and this kind of relationship with reference to the given function ‘f’ is an onto function or a surjection function. Thus the inverse function being an injunctive and a surjection function, is called a bijective function.

Let us consider a function f whose domain is the set X and the codomain is the set Y. The function f is invertible if there exists another function g whose domain is Y and its codomain is X. These two functions can be represented as f(x) = Y, and g(y) = X. For this situation, if the function f(x) is inverse, then its inverse function g(x) is unique.

If the composition of two functions f(x), and g(x), results in an identity function f(g(x))= x, then the two functions are said to be inverses of each other. If the application of a function to x as input gives n output of y, then the application of another function g to y should give back the value of x. Hence the inverse of a function reverses the function. The domain of the given function becomes the range of the inverse function, and the range of the given function becomes the domain of the inverse function.

## Steps To Find An Inverse Function

The following sequence of steps would help in conveniently finding the inverse of a function. Here we consider a function f(x) = ax + b, and aim at finding the inverse of this function through the following steps.

- For the given function f(x) = ax + b, replace f(x) = y, to obtain y = ax + b.
- Interchange the x with y and the y with x in the function y = ax + b to obtain x = ay + b.
- Here solve the expression x = ay + b for y. And we obtain y = (x – b/a
- Finlly replace y = f
^{-1}(x), and we have f^{-1}(x) = (x – b)/a.

## Graph of An Inverse Function

The injective function is the reflection of the origin function with reference to the line y = x, and is obtained by swapping (x, y) with the (y, x).

If the graphs of two functions are given, we can identify whether they are inverses of each other. If the graphs of both functions are symmetric with respect to the line y = x, then we say that the two functions are inverses of each other. This is because of the fact that if (x, y) lies on the function, then (y, x) lies on its inverse function.

## Examples of Inverse Function

**Example 1:** Find if the functions f(x) = 2x + 3 and g(x) = (x – 3) /2 are inverses of each other using the inverse function formula.

**Solution:**

We will verify whether (f o g)(x) = (g o f)(x) = x.

(f o g)(x) = f (g(x))

= f ((x-3)/2)

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

= x – 3 + 3

= x

(g o f)(x) = g (f(x))

= g (2x + 3)

= (2x + 3 – 3)/2

= 2x/2

= x

We proved that (f o g)(x) = (g o f)(x) = x.

By inverse function formula, f and g are inverses of each other.

**Answer:** We proved that f = g^{-1} and g = f^{-1}.

## FAQs on Inverse Function

### What Is An Inverse Function?

The inverse of a function f is denoted by f^{-1} and it exists only when f is both one-one and onto function. Note that f^{-1} is NOT the reciprocal of f. The composition of the function f and the reciprocal function f^{-1} gives the domain value of x.

(f o f^{-1}) (x) = (f^{-1} o f) (x) = x

### How Do You Find the Inverse of a Function?

The following sequence of steps help in finding the inverse of a function. Here we consider a function f(x) = ax + b.

- For the given function f(x) = ax + b, replace f(x) = y, to obtain y = ax + b.
- Interchange the x with y and the y with x in the function y = ax + b to obtain x = ay + b.
- Here solve the expression x = ay + b for y. And we obtain y = (x – b/a
- Finlly replace y = f
^{-1}(x), and we have f^{-1}(x) = (x – b)/a.

### What Is an Example of An Inverse Function?

The example of a inverse function is a function f(x) = 2x + 3, and its inverse function is f^{-1}(x) = (x – 3)/2.

### How To Find Inverse Function of Trigonometric Functions?

The inverse function of a trigonometric function is similar to finding the inverse of a normal function with algebraic expressions.

### What Is the Domain and Range of An Inverse Function?

The domain and range of an inverse function is obtained by swapping the domain and range of the given function. The domain of the given function becomes the range of the inverse function, and the range of the given function becomes the domain of the inverse function.

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