What is Inverse?
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 Inverse
1. What does the term 'inverse' mean in the context of mathematics?
In mathematics, an inverse refers to something that reverses or 'undoes' the effect of another operation or function. For example, subtraction is the inverse operation of addition because it reverses the action of adding. Similarly, an inverse function reverses the mapping of the original function, returning the original input value.
2. What are the inverse operations for basic arithmetic?
The basic inverse operations in arithmetic are paired to cancel each other out:
- The inverse operation of addition is subtraction. (e.g., 5 + 3 = 8, and 8 - 3 = 5)
- The inverse operation of multiplication is division. (e.g., 5 × 3 = 15, and 15 ÷ 3 = 5)
3. What is the difference between an additive inverse and a multiplicative inverse?
The main difference lies in the identity element they produce:
- An additive inverse of a number is what you add to it to get zero (the additive identity). For any number 'a', its additive inverse is '-a'. For example, the additive inverse of 7 is -7 because 7 + (-7) = 0.
- A multiplicative inverse (or reciprocal) of a number is what you multiply it by to get one (the multiplicative identity). For any non-zero number 'a', its multiplicative inverse is '1/a'. For example, the multiplicative inverse of 7 is 1/7 because 7 × (1/7) = 1.
4. How do you find the inverse of a function algebraically?
To find the inverse of a function, such as f(x), you can follow these steps:
- Replace f(x) with y.
- Swap the variables x and y in the equation.
- Solve the new equation for y.
- Replace this new y with the inverse function notation, f⁻¹(x).
For example, for f(x) = 2x - 3, swapping variables gives x = 2y - 3. Solving for y gives y = (x+3)/2. Thus, f⁻¹(x) = (x+3)/2.
5. What symbol is used to denote an inverse function?
The inverse of a function 'f' is denoted by f⁻¹. If we have a function f(x), its inverse is written as f⁻¹(x). It is important to note that the '-1' is not an exponent; f⁻¹(x) does not mean 1/f(x).
6. Why is the concept of an inverse function so important in solving equations?
Inverse functions are crucial for solving equations because they allow us to isolate variables. When a variable is trapped inside a function (like a logarithm, a trigonometric function, or an exponent), we can apply the corresponding inverse function to both sides of the equation to 'free' the variable and find its value.
7. Does every mathematical function have an inverse?
No, not every function has an inverse. For a function to have a well-defined inverse, it must be bijective, which means it is both one-to-one (each output is linked to only one input) and onto. In simpler terms, a function has an inverse if it passes the Horizontal Line Test, meaning any horizontal line drawn on its graph intersects the graph at most once.
8. What is the relationship between the graph of a function and its inverse?
The graph of an inverse function, f⁻¹(x), is a reflection of the graph of the original function, f(x), across the line y = x. This means that if the point (a, b) is on the graph of f(x), then the point (b, a) will be on the graph of its inverse, f⁻¹(x).
9. How are inverse trigonometric functions like sin⁻¹(x) different from reciprocals like 1/sin(x)?
This is a common point of confusion. They are fundamentally different concepts:
- Inverse Function (sin⁻¹(x) or arcsin(x)): This function answers the question, "What angle has a sine value of x?" It 'undoes' the sine function to find an angle.
- Reciprocal Function (1/sin(x) or csc(x)): This is the multiplicative inverse of the sine value. It is a value, not an angle.
In short, sin⁻¹(x) finds an angle, while 1/sin(x) calculates a ratio.
