# 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 = axn + b Form.

If f(x) = axn + b , Example:

f(x) = 3x4 + 8

Find the value of the f-1(56).

Solution:

y = 3x4 + 8

x = 3y4 + 8

x – 8 = 3y4

 y4 = 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.

Compound Functions

Function Questions with Solutions

RISE KNOWLEDGE

03/09/2018

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