Real-Life Examples of Inverse in Mathematics
Inverse Mathematics
Are you wondering to know the meaning of inverse in Mathematics? Inverse is nothing but the opposite arithmetic operation or reverse the effect of each other. For example, the inverse of addition is subtraction and the inverse of multiplication is division.
The Inverse of Addition is Subtraction
Addition moves Forward whereas Subtraction moves Backward. Let us understand with an example:
30 + 4 = 34 can be reversed by 34 - 4 = 30. (Back to where we started)
The Inverse of Subtraction is Addition
Let us understand inverse of subtraction with an example:
12 - 3 = 9 can be reversed by 9 + 3 = 12 (Back to where we started)
Additive Inverse
Additive inverse is what we add to the given number to get Zero.
Example:
The additive inverse of -3 is 3 because -3 + 3 = 0
Also, the additive inverse of + 8 is - 8 because 8 - 8 = 0
The Inverse of Multiplication is Division
Multiplication can be “undone” by Division. Let us understand with an example:
Example: 6 8 = 48 can be reversed by 488 = 6 (Back to where we started)
The Inverse of Division is Multiplication
Division can be “undone” by Multiplication. Let us understand with an example:
Example: 18 3 = 6 can be reversed by 6 3 = 18 (Back to where we started)
Multiplicative Inverse
Multiplicative inverse is with what we multiply a given number to get the result 1.
In other words, it is simply the reciprocal of a given number. Let us understand with an example:
The multiplicative inverse of 6 is 16, because 6 16 = 1.
But Division With 0 Cannot be Done.
We cannot divide any number by 0.
Example:
6 0 = 0 cannot be reversed by 0 0.
Inverse Function
An inverse function can be “undone” by another function. Let us understand with an example:
Here, the function 3x + 5 is represented in a flow diagram below:
x Multiply by 3 3x Add 5 3x + 5
The inverse function goes the other way:
y - 53 Divided By 3 Y - 5 Subtract 5 y
Therefore, the inverse of 3x+ 5 is y - 53.
Inverse of Function Representation
The inverse is generally represented with a little -1 after the function name as shown below:
f⁻¹ (y)
It can be said as f inverse of y
Hence, the inverse of f(x) = 3x + 5 is written as:
f⁻¹ (y) = y - 53
Important Thing To Note About Inverse of A Function
The most important thing about the inverse of function is that it takes us back where we started.
Here, the function f turns the mango into an orange.
Then, the inverse function f⁻¹ turns the orange back into the mango.
Let us understand with an example.
Inverse Function Example
Find the inverse of 5x + 5, when the value of x is 3,
Solution:
Given function: 5x + 5
Substituting the value of x in the above function, we get
53 + 5
15 + 5 = 20
Therefore, 5x + 5 = 20 ( when the value of x is 3).
Now, we will use the inverse on 20 using the formula discussed above .
f⁻¹ (20) = 20 - 55 = 3
And now you can see, we get the 3 back again.
This can be written in one line as:
f⁻¹ (f(20)) = 3
Hence, we can say applying a function f and then its inverse f⁻¹ give us the original value back.
f⁻¹ (f(x)) = x
Conclusion
In short, inverse means opposite in direction or position. The term ‘inverse’ is derived from the Latin word ‘inverses’ which means to turn upside down or inside and outside. In other words, an inverse operation is a mathematical operation that undoes what was performed on the previous operation.
FAQs on What Does Inverse Mean in Maths?
1. What does the term 'inverse' mean in mathematics?
In mathematics, an inverse is an element or an operation that 'reverses' or 'undoes' another element or operation. When an element is combined with its inverse, it yields the identity element for that operation. For example, adding a number and its inverse results in 0 (the additive identity), while multiplying a number by its inverse results in 1 (the multiplicative identity).
2. What is the difference between an additive inverse and a multiplicative inverse?
The key difference lies in the operation they 'undo'.
An additive inverse of a number 'x' is the number that, when added to 'x', gives zero. It is simply the negative of the number. For example, the additive inverse of 7 is -7, because 7 + (-7) = 0.
A multiplicative inverse (or reciprocal) of a number 'x' is the number that, when multiplied by 'x', gives one. For example, the multiplicative inverse of 7 is 1/7, because 7 * (1/7) = 1.
3. How do you find the inverse of a function, for example, f(x) = 2x + 3?
To find the inverse of a function, you swap the roles of 'x' and 'y' and then solve for the new 'y'. Let's find the inverse of f(x) = 2x + 3:
Step 1: Replace f(x) with y. So, y = 2x + 3.
Step 2: Swap x and y in the equation. So, x = 2y + 3.
Step 3: Solve for the new y.
x - 3 = 2y
y = (x - 3) / 2Step 4: Replace y with the inverse function notation, f⁻¹(x). The inverse function is f⁻¹(x) = (x - 3) / 2.
4. What is a real-world example of an inverse relationship in mathematics?
A great real-world example is the relationship between speed and time when travelling a fixed distance. If you need to travel 100 km:
If you increase your speed, the time it takes to cover the distance decreases.
If you decrease your speed, the time it takes increases.
This is an inverse relationship, where one quantity goes up as the other goes down, following the formula: Time = Distance / Speed.
5. How is the concept of an inverse function different from the inverse of a matrix?
While both concepts involve 'undoing' something, they apply to different mathematical objects and operations.
An inverse function, denoted as f⁻¹, reverses the mapping of a function. If f(a) = b, then f⁻¹(b) = a. It applies to functions and the operation is function composition.
An inverse of a matrix, denoted as A⁻¹, is another matrix that, when multiplied by the original matrix A, results in the identity matrix (I). It applies to square matrices and the operation is matrix multiplication.
So, one reverses an input-output process, while the other reverses a matrix transformation in linear algebra.
6. Can every mathematical operation or function have an inverse?
No, not every operation or function has an inverse. For an inverse to exist, certain conditions must be met.
For numbers, the number zero does not have a multiplicative inverse because you cannot divide by zero to get 1.
For functions, a function must be bijective (both one-to-one and onto) to have an inverse. A function like f(x) = x² does not have a general inverse because, for example, both f(2) and f(-2) equal 4. The inverse function wouldn't know whether to map 4 back to 2 or -2.
7. Does the concept of 'inverse' exist in geometry?
Yes, the concept of inverse is fundamental in geometry, particularly in transformations. A geometric transformation's inverse is the transformation that maps the image back to its original position. For example:
The inverse of a translation (sliding a shape) is a translation in the opposite direction.
The inverse of a rotation by an angle θ is a rotation by -θ around the same point.
A reflection is its own inverse. If you reflect a shape across a line and then reflect it again across the same line, it returns to its original position.
