Inverse Functions

When we talk about inverse functions or anti functions, we’re talking about functions that can be reversed into another function. To put it another way, if any function “f” transforms x into y, then the inverse of that function will transform y into x. If the function is denoted by the letters ‘f’ or ‘F,’ then the inverse function is denoted by the letters f-1 or F-1, respectively. It is important not to confuse (-1) with the exponent or the reciprocal in this context. 

Definition

A function takes in input values, executes specific operations on those values, and generates an output in response. Assuming that the resultant is correct and that the inverse function operates, it returns to the original function.

The inverse function returns the value that was passed into the function that produced the output.

When it comes to functions, f and g are inverses of one another; f(g(x)) = g(f(x)) = x. A function that is made up of its inverse returns the value that was originally entered. 

Example: f(x) = 2x + 5 = y 

Then, g(y) = (y-5)/2 = x is the inverse of f(x). 

Note

• It is the relationship that is formed when the independent variable is exchanged with the variable that is dependent on a specific equation, and this inverse relationship may or may not be a function

• If the inverse of a function is the function itself, it is referred to as the inverse function, which is symbolised by the symbol f-1 (x)

Inverse function graph

Over the line y=x, the graph of the inverse of a function reflects two things: first, it depicts the function and second, it depicts its inverse, which is the function’s inverse. Slope value 1 is assigned to this line in the graph since it passes through the origin. It can be expressed as follows: 

y = f-1(x) 

which is equal to;

x = f(y) 

In some ways, this relationship is comparable to the relationship y = f(x), which defines the graph of f, except that the parts of x and y are flipped here. Consequently, in order to show the graph of f-1, we must swap the positions of x and y in the coordinate plane. 

Finding the inverse of a function

In most cases, the process of determining an inverse involves switching the coordinates of the two variables. This freshly constructed inverse is a relation, not necessarily a function because it is a relation.

To ensure that the inverse of the original function is likewise a function, the original function must be a one-to-one function. In order for a function to be considered one to one, every second element must correspond to the same initial value as before (values of x and y are used only once). 

In order to determine whether a function is one-to-one, you can use the horizontal line test to determine its value. It is possible for a function to be one-to-one if a horizontal line intersects the original function in a single region. In this case, the inverse is also a function. 

Types of inverse functions

There are many different types of inverse functions, including the inverse of trigonometric functions, rational functions, hyperbolic functions, and log functions, to name a few. The inverses of some of the most frequently encountered functions are listed below. 

Inverse Trigonometric functions

The inverse trigonometric functions, commonly known as arc functions, are used to calculate the length of the arc that must be drawn in order to acquire a given value. It is possible to compute six inverse trigonometric functions, which are the arcsine (sin-1), the arccosine (cos-1), arctangent (tan-1), arcsecant (sec-1), arccosecant (cosec-1), and the arccotangent (cot-1). 

Inverse Rational function: 

A rational function is a function of the type f(x) = P(x)/Q(x), where P(x) and Q(x) are both positive integers. The instructions below will guide you through the process of determining the inverse of a rational function. An example is also provided below to assist you in better comprehending the idea in question. 

• Step 1: Replace f(x) = y

• Step 2: Interchange x and y

• Step 3: Solve for y in terms of x

• Step 4: Replace y with f-1(x) and the inverse of the function is obtained. 

Inverse Hyperbolic functions

In the same way that inverse trigonometric functions are the inverses of trigonometric functions, inverse hyperbolic functions are the inverses of hyperbolic functions. The inverse hyperbolic functions are sinh-1, cosh-1, tanh-1, csch-1, coth-1, and sech-1. 

Inverse Trigonometric ratios formula

There are just a few fundamental formulas for inverse trigonometric ratios, but by using trigonometric functions, we can derive many more inverse trigonometric ratio formulas than those listed below. In the following section, we will look at some of the inverse trigonometric ratio formulae that are connected to the inverse trigonometric functions. 

• Sin-1(-x) = -Sin-1x
• Tan-1(-x) = -Tan-1x
• Cosec-1(-x) = -Cosec-1x
• Cos-1(-x) = π – Cos-1x
• Sec-1(-x) = π – Sec-1x
• Cot-1(-x) = π – Cot-1x 

Inverse functions examples

Example: Find the inverse of the function f(x) = (3x+2)/ (x-1).

Solution

First, replace f(x) with y and the function becomes,

Y = (3x+2)/(x-1)

By replacing x with y we get,

X = (3y+2)/(y-1)

Now, solve y in terms of x :

X (y – 1) = 3y + 2

• Xy – x = 3y +2

• Xy – 3y = 2 + x

• Y (x – 3) = 2 + x

• Y = (2 + x) / (x – 3)

So, y = f-1(x) = (x+2)/(x-3)

Conclusion

The following are just a few examples of how Inverse trigonometric ratios might be used: The measure of the unknown angles of a right-angled triangle can be determined using this formula. The angle of depth or angle of inclination measurements is made with this instrument. A bridge’s angle and the angle of the supports are calculated using this formula by architects. Carpenters use this method to achieve the desired cut angle.