An **inverse function** or an anti function is defined as a function, which can reverse another function. In simple words, if any function “f” takes x to y then, the inverse of “f” i.e. “f^{-1}” will take y to x. We will cover here all the topics regarding this function along with examples, such as;

- Definition
- How to Find the Inverse of a Function?
- Types
- Inverse of Trigonometric Functions
- Inverse of Rational Functions
- Inverse of Hyperbolic Functions
- Inverse Logarithmic Functions and Inverse Exponential Functions

- Example Questions from Inverse of Function

## Inverse Function Definition

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

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.

**Note:**

- The relation, developed when the independent variable is interchanged with the variable which is dependent in 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).

## 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 be also 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 Functions

There are various types of an inverse function like the inverse of trigonometric functions, rational functions, hyperbolic functions and log functions. These are discussed in detail below. The inverses of some of the most common functions are given below.

Function | Inverse 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). To learn more about the inverse trigonometric functions along with their graphs, follow the linked article.

### Inverse Rational Function

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

**Example:**

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)

### Inverse Hyperbolic Functions

Just like the inverse trigonometric function, the inverse hyperbolic functions are the inverses of the hyperbolic functions. There are 6 main inverse hyperbolic functions 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 function 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 abilities.

**Example: **

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}

### Inverse Functions Examples

**Question: To solve an equation: f(x) = 2x + 3, at x = 4**

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(f(4)) = f(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 of 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 |

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