Inverse Functions

Inverse Function Definition

If I ask you a question “What is an inverse function?” What answer do we have? Let us see. Inverse functions are functions that can inverse other functions. It is just like undoing another function that leaves you to where you started. If a function is to drive from home to the shop then the inverse function will be to drive from the shop to back home. It is very much like a game of “doing” and “undoing”. A function starts with a value then performs some operation on it and the created output leads to the answer. The inverse function starts with the output answer then performs some operation on it and brings us back to the starting value. An inverse function basically interchanges the first and second elements of each pair of the original function.

For example, consider that a graph of a function has (a and b) as its points, the graph of an inverse function will have the points (b and a ). An inverse function is written as f\[^{-1}\](x)

Let us take another example, consider f(x) = 3x – 6. What happens to x? We first multiply by 3 and then subtract 6 from the result. But in the reverse function, we follow the steps backward by first adding 6 to undo the subtraction and then divide it by 3 to undo the multiplication.  

How to find the Inverse of a Function

Since we now know what an inverse function is, wouldn’t you want to know how to solve inverse functions? What are we waiting for then? Let’s unwrap the mystery.

There are three methods to find the inverse of a function.

  1. Simply swapping the ordered pairs

  2. Solve it algebraically

  3. Using A graph

Finding Inverse By Swapping: As the name suggests, we just need to swap the values of x and y. 

Examples Time:

Example 1) Find the inverse function if f(x) = {(3,4)(1,-2)(5,-1)(0,2)}

Solution 1) Since the values x and y are used only once, the function and the inverse function is a one-to-one function. Therefore, the inverse function will be:

f\[^{-1}\](x) = {(4,3)(-2,1)(-1,5)(2,0)}

Finding Inverse Algebraically:  To find inverse algebraically we have to follow three steps:

Step 1) Set the function as y

Step 2) Swap the variables x and y

Step 3) Solve y

Example 1) f(x) = x - 4

Solution 1) y = x - 4        (step 1)

                  x = y - 4         (step 2)

                  x + 4 = y        (step 3)

                 f\[^{-1}\](x) = x + 4 (one-to-one function)


Finding Inverse Using Graph: The graph of an inverse function is the reflection of the original graph over the identity line y = x. 

Example 1) Graph the inverse function of y = 2x + 3

Consider the original function as y = 2x + 3 which is drawn in blue. If we reflect it over the identity line that is y = x, the original function will become the red dotted line on the graph. The red straight dotted line passes the vertical line test for functions. The inverse function of y = 2x + 3 is also a function. 

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Types of Inverse Function

There are different types of inverse functions like the inverse of trigonometric functions, the inverse rational functions, inverse hyperbolic functions, and inverse log functions. 

Inverse Trigonometric Functions

We can also call the inverse trigonometric functions as arc functions because they produce the length of the arc which is necessary to obtain that particular value. There are six inverse trigonometric functions which are named as:

  1. arcsine (sin\[^{-1}\]),

  2. arccosine (cos\[^{-1}\]),

  3. arctangent (tan\[^{-1}\]), 

  4. arcsecant (sec\[^{-1}\]), 

  5. arccosecant (cosec\[^{-1}\]), 

  6. arccotangent (cot\[^{-1}\]).

Inverse Rational Function

A rational number is a number which can be written as f(x) = P(x)/Q(x) where Q(x) is ≠ 0. In order to find the inverse function of a rational number, we have to follow the following steps. 

Step 1: first we have to replace f(x) = y

Step 2: Then interchange the values x and y

Step 3: In this step, we have to solve for y in terms of x

Step 4: Finally we have to replace y with f\[^{-1}\](x) and thus we can obtain the inverse of the function.

Inverse Hyperbolic Functions

Just like the inverse trigonometric function, in the same way, the inverse hyperbolic functions are the inverses of the hyperbolic functions. The 6 main inverse hyperbolic functions are:

  1. sinh\[^{-1}\] 

  2. cosh\[^{-1}\] 

  3. tanh\[^{-1}\] 

  4. csch\[^{-1}\] 

  5. coth\[^{-1}\] 

  6. sech\[^{-1}\] 

Inverse Logarithmic Functions and Inverse Exponential Function

The natural logarithm functions are inverse of the exponential functions. 

Inverse Function Examples and Solutions

Example 1)  Find the Inverse Function

x

1

-2

-1

0

2

3

4

-3

f(x)

2

0

3

-1

1

-2

5

1


Solution 1) Since the value of 1 is repeated twice, the function and the inverse function are not one-to-one function. Therefore, after swapping the values, the inverse function will be: 

 f\[^{-1}\](x) = {(2,1)(0,-2)(3,-1)(-1,0)(1,2)(-2,3)(5,4)(1,-3)}

Example 2) Find the function f(x) if the inverse function is given as  f\[^{-1}\](x) = - \[\frac{1}{2}\]x+1

Solution 2) At first look the question might seem a different type of problem but it is not. It can be solved in the same way as example 1 using the same steps.

y = - \[\frac{1}{2}\]x+1

x =  - \[\frac{1}{2}\]y+1

x - 1 = - \[\frac{1}{2}\]y

-2(x-1) = y

f(x) = y = -2x + 2

FAQ (Frequently Asked Questions)

Q1. Is Reciprocal and Inverse the Same? 

Ans. A reciprocal can be an inverse but an inverse cannot be reciprocal. A reciprocal is a multiplicative inverse. Basically an inverse function undoes the original function by switching the input and output. For example, the reciprocal of x = y + 2 will be x = 1/ y+4 whereas its inverse will be y = x - 2. Also a reciprocal can be represented in different ways but does not have any specific sign whereas an inverse is represented as f-1(x).