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


InverseFnc1


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


InverseFnc2


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 ,

InverseFnc3


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



InverseFnc4


f-1(56) = 2


Graphics Of The Inverse Functions

InverseFnc5


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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