# Inverse Functions

## Functions in mathematics. Concept of inverse function. Finding the inverse of a function. Subject expression and solved questions.

A and B are two not empty cluster.

Suppose, f(x) is a function that defined to the cluster "B" from the cluster "A".

If

f(x) = y, then is

f^{-1 }(y) = x

If a f(x) function matches up elements of a set "A" with elements of a "B" set, the inverse of this function matches up the elements of set “B” with the elements of set "A".

For to be inverse of a function, the function must be bijective.

**Example:**

Let we find value of the f(x) in the following for x = 2.

f(x) = 2x + 8

f(2) = 2•2 + 8

f(2) = 12

It is y = 12, for x = 2.

Now, let we find the inverse of the f(x) function. For this, let we write y instead of x, x instead of y. As following,

y = f(x)

y = 2x + 8

x = 2y + 8

2y = x – 8

y = | x – 8 | |

2 |

f^{-1}(x) = | x – 8 | |

2 |

Now, let we write 12 instead of x.

f^{-1 }(12) = | 12 – 8 | |

2 |

= 2

As it is seen,

If f: A → B, f^{-1}: B → A

If f(2) = 12, f^{-1} (12) = 2

**Finding The Inverse Of a Function**

First, the function is written as follows,

y = f(x)

Then, it is written y instead of x, x instead of y.

**1. Functions In ax + b Form**

The inverse of a function that in the form ax + b is,

f-1(x) = | x – b | |

a |

**Example:**

f(x) = 12x + 5

Find the function f^{-1 }(x)

**Solution:**

y = f(x)

y = 12x + 5

Let we write y instead of x, x instead of y.

x = 12y + 5

x – 5 = 12y

y = | x – 5 | |

12 |

f^{-1}(x) = | x – 5 | |

12 |

**2. Functions In y = (ax + b)/c Form**

If

f(x) = | ax + b | |

c |

f^{-1}(x) = | cx – b | |

a |

**Example:**

f(x) = | 15x – 7 | |

4 |

Find the function f^{-1}(x).

**Solution:**

y = | 15x – 7 | |

4 |

x = | 15y – 7 | |

4 |

4x = 15y – 7

4x + 7 = 15y

y = | 4x + 7 | |

15 |

**3. Functions In y = (ax + b)/(cx + d) Form**

**Example:**

f(x) = | 16x + 5 | |

4x +6 |

Find the inverse of the function f(x).

**Solution:**

y = | 16x + 5 | |

4x +6 |

We write x instead of y, y instead of x.

x = | 16y + 5 | |

4y +6 |

x• (4y + 6) = 16y + 5

4xy + 6x = 16y + 5

16y – 4xy = 6x – 5

y(16 – 4x) = 6x – 5

y = | 6x - 5 | |

–4x +16 |

y = | –6x + 5 | |

4x – 16 |

**Example:**

f(x) = | 5x – 3 | |

9x – 6 |

Find the inverse of the function f(x).

**Solution:**

y = | 5x – 3 | |

9x – 6 |

We write x in place of y, y in place of x.

x = | 5y – 3 | |

9y – 6 |

x(9y – 6) = 5y – 3

9xy – 6x = 5y – 3

9xy – 5y = 6x – 3

y(9x – 5) = 6x – 3

y = | 6x – 3 | + |

9x – 5 |

**4. The inverse of the inverse of a function is equal to that function.**

[f^{-1}(x)]^{-1} = f(x)

**Example:**

f^{-1 }(x) = | 5x – 8 | |

12 |

Find the value of the f(6).

**Solution:**

For find the function f(x), we must find the inverse of the f^{-1}(x).

For this, in the f^{-1 }(x) function we write y instead of x, x instead of y.

y = | 5x – 8 | |

12 |

x = | 5y – 8 | |

12 |

12x = 5y – 8

5y =12x + 8

y = | 12x + 8 | |

5 |

f(x) = | 12x + 8 | |

5 |

f(6) = | 12•6 + 8 | |

5 |

f(6) = 16

Let’s now write this result in place of x in the function f^{-1}(x).

y = | 5x – 8 | |

12 |

y = | 5•16 – 8 | |

12 |

y = 6

So, if f(16) = 6, f^{-1}(6) = 16

**5. Functions In y = ax ^{n} + b Form.**

If f(x) = ax^{n} + b ,

**Example:**

f(x) = 3x^{4} + 8

Find the value of the f^{-1}(56).

**Solution:**

y = 3x^{4} + 8

x = 3y^{4} + 8

x – 8 = 3y^{4}

y^{4} = | x – 8 | |

3 |

f-1(56) = 2

**Graphics Of The Inverse Functions**

The red colored graph in the figure above belongs to the function f(x).

As can seen in the figure, it is being y = 5 for f(8). The blue colored graph is the inverse of the f(x), and it is being f^{-1}(5) = 8.

When drawing graph of inverse function, the y-axis values of the main function is written to the x – axis in the inverse function. The x-axis values of the main function is written to the y-axis in the inverse function.

**Function Questions with Solutions**

RISE KNOWLEDGE

03/09/2018

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