Have you ever come across a mathematical equation that changes a number from A to B and vice versa from B to A? This is called an inverse function of class 10. The inverse itself means a function that is opposite to the original function. This mathematical material is interesting to discuss.

The inverse is useful for finding relationships between certain numbers. Considering that this is a compulsory subject that you will definitely encounter in school, make sure you know the formula and typical questions related to reciprocals. This function can be used to change the equation to find the values of x and y.

Are you looking to flip learning? For this reason, let’s take a closer look at the following discussion!

## Inverted understanding

An inverse is a function used in mathematics to best describe the relationship between two mathematical operations. This is why this function is also called the inverse function.

If there are arithmetic operations A and B, then the inverse of A is B. The function F changes to F-1 if F is a binary function. A function is said to be binary when the domain and the code domain have the same number of members.

## Understanding binary and odd functions

In the opposite direction there are functions known as binary functions and odd functions. Both are used to indicate an inverse case.

A binary function itself is a function that occurs when the domain and the code domain have the same number of members. A function F becomes F-1 if F is a binary function. So, if there is an operation to calculate A and B, then the inverse of A is B.

Meanwhile, a one-to-one function is a function where each field is associated with a different cryptographic field. Another name for this function is injection function. An example is the function f(x) = x^{2}. This is called a one-to-one function because each domain x is mapped to a different code domain, x^{2}.

## Understand the inverse composition function

If you already understand the basic inverse function, also understand inverse composition so that it will be easier to calculate the inverse. Composite function means a joint function between two functions.

Writing is represented by the symbol “o” or read as a roundabout or formation. If there is an equation (fog), the way to read it is the rotating function g, which means that the function g is done first.

On the other hand, if there is (gof), it means that the rotor function f and the function f are completed first. The nature of job composition is as follows:

- It has an element of identity
- It is relational, not reciprocal

## Inverse symbol

Basically, an inverse is the opposite function. In mathematical calculations, it is written as the power minus one (^{-1}) of the corresponding function letters. To make it more clear, try looking at the following symbols:

- X=Y-1 (f
^{-1}) - y = x-1(x
^{-1})

The inverse writing example above shows that matrices X and Y are inverse or inverse matrices.

## Inverse function formula

Inverse has a very simple function. You can write this relationship in the following form:

**(and⁻¹)⁻¹ = and**

However, more complex mathematical functions cannot simply use the above formula. The following are commonly used function formulas:

## How to determine the inverse of a function

When calculating the relationship between two arithmetic operations, you must first determine the inverse of the function you want to calculate. it’s easy. There is a quick method you can try. But before that, understand the steps as follows:

Count f(x) = y

- In the equation above, this means that the variable f(x) is the variable y and that f(x) contains the variable x.
- Enter the equation x into the variable y.
- Enter the variable x into f-1(x) and the variable y into x.

This is the basis for using the inverse form in the equation.

## How to determine the inverse of a matrix

Before that, you first need to understand that the inverse of a matrix is the inverse of a matrix. This inverse result is obtained when the matrix is multiplied by the original matrix, so that it becomes the unit matrix.

The so-called identity matrix is a matrix whose diagonal elements are 1 (one), while the other elements are 0 (zero). To determine the inverse of a matrix, you can calculate it using a 2×2 or 3×3 reference. Here is an example of calculating the inverse matrix:

### 1. Urdu 2×2

### 2. Urdu 3×3

To get results like the ones mentioned above, here are the steps you need to do:

- Replace the matrix members on the main diagonal.
- Add a minus sign (-) for members not on the main diagonal line.
- Finally, partition each array element and its determinant.

## Examples of inverse function questions

Just looking at the formula may not be very helpful in understanding this function. Therefore, this article provides a set of sample questions that can be used as practice to become more familiar with inverses. Let’s start counting!

### 1. Question 1

If f(x) = x – 4, then f-1(x) is….

Solution | F(x) = y F(x) = x – 4 y = x – 4 x = y + 4 |

Next, replace the value of x with f-1(x) and the value of y with y + 4. Then, you get the answer f-1(x) = x + 4.

### 2. Question 2

If f(x) = 6 – 6x then f-1(x) = …

Solution | F(x) = y And (x) = 6 – 6x y = 6 – 6x 6x = 6 – y x = (6-y)/6 |

Now, replace the values of x and y respectively into f-1(x) = = 1 – 1/6x. So the answer is 1 – 1/6x.

### 3. Question 3

If f(x) = 2x + 1 then f-1(2) =…

### 4. Question 4

If f(x) = x^{2 }– 2 Then f-1(x) is…

- √x + 2
- √x + 8
- x2 – 2
- x2 – 8
- x2 + 2

Solution | Find the value of f-1(x) y = x ^{2} -2 s ^{2} = y + 2 x = √y + 2 f-1(x) = √x+2 |

### 5. Question 5

If f(x) = 3 + √x+3, then f-1(x) is…

- (S – 3)
^{2}– 3 - -(S – 3)
^{2}+ 1 - (S – 3)
^{2}+ 9 - (X-9)
^{2}-1 - (x + 9)
^{2}– 3

Solution | Finding the value of F-1(x)
√x + 3 = y – 3 |

### 6. Question 6

It is known that the function f(x) = 3x – 5. Determine the inverse of this linear function!

Solution | Find the value of f-1(x)
F(x) = 3x – 5 |

Then the inverse of the function f(x) = 3x – 5 is f-1(x) = 1/3 (y + 5)

### 7. Question 7

If the function has the form f(x) = 2x – 1 and f-1(x) is the inverse of f(x). What is the value of f-1(7)?

Solution | Looking for value
F(x) = 2x – 1 Finding the value of f-1(7) |

The answer to the inverse function question f-1(7) is 4.

### 8. Question 8

Calculate the inverse of the formula of the function f(x) = (x + 3)/(x – 4)…

Solution | Find the value of f-1(x) y = (x + 3)(x – 4) y(x – 3) = x + 4 yy – 3y = x + 4 yy – x = 3y + 4 x(y – 1) = 3y + 4 x = (3y + 4)/(y – 1) |

After you have the formula, all you have to do is change the position of x and y to fˉ¹(x) and xf‾¹(x) = (3x + 4)/(x – 1). Then the final result is f-1(x) = (3x + 4)/(x – 1).

### 9. Question 9

Given the formula f(x)=(4x+2)/(3x-1). Find the inverse of the formula…

Solution | Find the value of f-1(x) y = f(x) Y = (4x+2)/(3x-1) Y(3x-1) = 4x+2 3xy – y = 4x+2 3xy-4x = 2+y x(3y-4) = 2+y x = (2+y)/3y-4 |

After swapping positions, this calculation results in the inverse of f(x)=(4x+2)/(3x-1) being f-1(x) = (2+x)/(3y-4).

### 10. Question 10

Given the function f(x) = 3x – 5. Determine the inverse and get the value of f-1(x)…

Solution | Find the value of f-1(x) y = 3x – 5 f-1(x) = 3f-1(x) – 5 f-1(x) – 3f-1(x) = -5 f-1(x).(1 – 3) = -5 f-1(x) = -5/-2 f-1(x) = 5/2 |

Then the inverse of the rational inverse function f(x) = 3x – 5 is 5/2 or 2.5.

### 11. Question 11

The equation has the function f(x) = 2x – 3. Calculate f-1(5)…

Solution | Find the value of xy = 2x – 3y + 3 = 2x x = (y + 3)/2 Finding the value of f-1(5) |

Then f-1(5) for the function f(x) = 2x – 3 is 4.

### 12. Question 12

When f(x) = 8 – 1/2x, what is the inverse of the function?

Solution | Find the value of y
And (x) = 8 – 1/2x Find the value of f-1(x) |

In conclusion, the inverse of f(x) = 8 – ½x is f-1(x) = -2(y – 8).

Use the set of inverse function questions and how to solve them above to practice until you become more adept at working with this mathematical formula. With a little practice, you’ll definitely have no problem doing the opposite! The key is to stay consistent and keep training.

## Inverse function practice questions

After seeing some examples above, please try to calculate the inverse in your own way. Here are 10 practice questions, be sure to do them yourself before looking at the answer key below:

- Find the inverse of f(x) = 3x + 2!
- If f(x) = 4x – 5, find the inverse formula f-1(9)!
- Calculate the function f(x) = 5x – 3 to find the inverse!
- What is the inverse of f(x) = (2x + 3)/(x – 1)?

Answer key

f-1(x) = 3x + 2

y = 3x + 2

Y – 2 = 3x

x = (y – 2)/3

Replace f(x) with y to become the equation:

y = 4x – 5

Y + 5 = 4x

x = (y + 5)/4

And -1(9) = (9 + 5)/4 = 14/4 = 3.5

y = 5x – 3

y + 3 = 5x

x = (y + 3)/5

F-1(x) = (x + 3)/5

y = (2x + 3)/(x – 1)

y(x – 1) = 2x + 3

Yx – y = 2x + 3

yy – 2x = y + 3

x(y – 2) = y + 3

x = (y + 3)/(y – 2)

F-1(x) = (x + 3)/(x – 2)