Definition Formula and Examples of Inverse in Maths
In general inverse meaning is “something that is opposite or reverses”. When we define inverse meaning we will talk mainly from the Mathematics point of view where we find the inverse of various operations and functions.
What is the Meaning of Inverse in Mathematics?
The inverse meaning in Math is “a function or operation which reverses the order or operation of another function or operation”.
Inverse Math Example: The inverse operation of addition is subtraction, the inverse operation of multiplication is division.
Now let us look into basic concepts of various operations and their inverse operation to understand the inverse Math definition clearly.
The Inverse of a Number
Consider a number x where x is not equal to zero. Then the inverse of a number x is the reciprocal of the number that is 1/x.
Ex: Inverse of a number 100 is 1/100, the inverse of a number 34 is 1/34.
The Inverse of Various Operations
In this section, we will clear our doubts about inverse meaning by applying the inverse to the basic operations of Mathematics.
Inverse Operation in Addition
The addition is one of the foremost operations in arithmetic where we add the numbers to find the total or sum of the numbers.
So, the inverse of the addition operation is subtraction. That is whenever we are asked to find the inverse of addition we have to subtract.
Ex: The addition of two numbers 25 and 15 can be written as 25 + 15 = 40. So, the inverse of this addition operation will be 40 - 15 = 25.
Inverse Operation on Subtraction
Subtraction is an operation where we remove numbers from the collection.
The inverse of the subtraction operation is addition.
Ex: The subtraction of two numbers 40 and 30 can be written as 40-30 = 10. So, the inverse operation of this will be 10 + 30 = 40.
Inverse Operation on Multiplication
Multiplication is an operation where we combine groups of equal sizes. Multiplication is nothing but a repeated addition process.
So, the division is the inverse of multiplication.
Ex: The multiplication of two numbers 4 and 6 is 4 × 6 = 24. The inverse operation of this is a division which is as follows: 24/6 = 4.
Inverse Operation on Division
The division is an operation where we divide the group of things into parts.
The inverse operation of division is multiplication.
Ex: The division of a number 45 by 5 is 45/5 = 9. The inverse operation of this is multiplication which is as follows: 9 × 5 = 45.
Additive Inverse
The Additive inverse of a number is the value that results in a sum as zero when added with the original value.
Ex: The additive inverse of +7 is -7. So, the sum will be +7 - 7 = 0.
Multiplicative Inverse
The Multiplicative inverse of a number is the value that results as one when multiplied with the original value.
Ex: The multiplicative inverse of 6 is 1/6. So, when we multiply these 2 values we get (6 × 1)/6 = 1.
The Inverse of a Function
The inverse function is a function that reverses the other function's action.
Ex: Consider the function f(x) = 7x + 2 = y. So, the inverse function will be g(x) = (y - 2)/7 = x.
So, the inverse function of 7x + 2 is (y - 2)/7.
The Inverse of a Trigonometric Function
The trigonometric functions are the functions that relate the right-angled triangle angle with the ratios of the side of the triangle.
Ex: sin θ = Opposite side/Hypotenuse. So, we find the angle of the triangle by using this formula. What if we have to calculate the hypotenuse of a triangle if an angle is given. Then we will use the inverse function to calculate the hypotenuse. So, the inverse is written as
θ = sin-1(Opposite side/Hypotenuse) is the inverse of the trigonometric sine function.
Similarly, cos θ = Base/Hypotenuse. So, inverse is θ = cos-1(Base/Hypotenuse).
Tan θ = Opposite side/ Base. So, inverse is θ = tan-1 (Opposite side/ Base).
The Inverse of an Exponential Function
The inverse of an exponential function is a logarithmic function.
Ex: Consider an exponential function 43 = 64. So, the inverse of this function will be a logarithmic function log4 (64) = 3.
Problems on Inverse
1. Find the inverse of a number 4, 14, 25 and 36.
Ans: To find the inverse of a number, we have to take the reciprocal of the given numbers.
So, the inverse of a number 4 will be 1/4.
The inverse of a number 14 will be 1/14.
The inverse of a number 25 will be 1/25.
The inverse of a number 36 will be 1/36.
2. Find the inverse of the addition of 2 numbers 35 and 74.
Ans: The inversion of addition is subtraction. The sum of the numbers 35+74 = 109. So the inverse will be 109 - 74 = 35.
3. Find the inverse of the multiplication of the two numbers 5 and 9.
Ans: The inverse of the multiplication is division. So the product of the two numbers is 5 × 9 = 45. Therefore the inverse of the two numbers is 45/9 = 5.
4. Find the additive inverse of the following numbers. -5, -2, 5, 10.
Ans: The additive inverse 0f -5 is +5 because -5+5 = 0 which proves the additive inverse property.
The additive inverse of -2 is +2 because -2+2 = 0.
The additive inverse of 5 is -5 because +5-5 = 0.
The additive inverse of 10 is -10 because +10-10 = 0.
5. Find the inverse of the function 10y - 3.
Ans: To find the inverse of the given function consider f(y) = 10y - 3 = x. So, the inverse of this function will be (x + 3)/10.
6. Find the inverse of the exponential function 63.
Ans: We know that 63 = 216. So, the inverse of this exponential function is log6 216 = 3.
Conclusion
The inverse operation of solving equations is important because it allows the reversal of Mathematical operations. One of the most important questions, once the Mathematical procedure is introduced, is how to reverse it.
A function can be viewed as mapping things of one type to things of a different type. The opposite of the function indicates how the original value is returned. Without really thinking about that, we do a lot of inverse operations in everyday life.
FAQs on Understanding Inverse in Mathematics
1. What is an inverse in mathematics?
An inverse in mathematics is a value or operation that reverses the effect of another value or operation. For example:
- The additive inverse of 5 is −5 because 5 + (−5) = 0.
- The multiplicative inverse of 4 is 1/4 because 4 × 1/4 = 1.
- An inverse function reverses the original function’s output back to its input.
2. What is the multiplicative inverse?
The multiplicative inverse of a number is the value that gives 1 when multiplied by the original number. For any nonzero number a, its multiplicative inverse is 1/a.
- Example: The inverse of 8 is 1/8.
- Example: The inverse of −3 is −1/3.
3. What is the additive inverse?
The additive inverse of a number is the value that gives 0 when added to the original number. For any number a, its additive inverse is −a.
- Example: The additive inverse of 7 is −7.
- Example: The additive inverse of −10 is 10.
4. What is an inverse function?
An inverse function is a function that reverses the effect of another function. If f(x) turns x into y, then its inverse f⁻¹(x) turns y back into x.
- If f(x) = 2x + 3, then solving for x gives f⁻¹(x) = (x − 3)/2.
- Applying both gives: f(f⁻¹(x)) = x.
5. How do you find the inverse of a function step by step?
To find the inverse of a function, interchange x and y and solve for y. Follow these steps:
- 1. Write the function as y = f(x).
- 2. Swap x and y.
- 3. Solve for y.
- 4. Replace y with f⁻¹(x).
- Swap: x = 3y − 4
- Solve: y = (x + 4)/3
- Inverse: f⁻¹(x) = (x + 4)/3
6. What is the inverse of a matrix?
The inverse of a matrix A is a matrix A⁻¹ such that A × A⁻¹ = I, where I is the identity matrix. For a 2×2 matrix:
A = [a b; c d]
The inverse is:
A⁻¹ = (1/(ad − bc)) [d −b; −c a]
- The determinant (ad − bc) must not be 0.
7. What is the inverse of a fraction?
The inverse of a fraction is found by swapping its numerator and denominator. This is also called the reciprocal.
- The inverse of 3/5 is 5/3.
- The inverse of −7/2 is −2/7.
8. What is the difference between inverse and reciprocal?
The term reciprocal usually refers specifically to the multiplicative inverse of a number, while inverse is a broader term used in different mathematical contexts.
- Reciprocal: 5 → 1/5.
- Additive inverse: 5 → −5.
- Inverse function: reverses another function.
9. When does an inverse function exist?
An inverse function exists only if the original function is one-to-one (injective). This means each output corresponds to exactly one input.
- Graphically, it must pass the horizontal line test.
- If it fails, you may need to restrict the domain.
10. What is the inverse of an exponential function?
The inverse of an exponential function is a logarithmic function. If f(x) = aˣ, then its inverse is f⁻¹(x) = logₐ(x).
- Example: If f(x) = 2ˣ, then f⁻¹(x) = log₂(x).
- Because 2³ = 8, we have log₂(8) = 3.
