Inverse Function

An inverse function is an “undo” function. A function which is just the opposite of the original input. To graph an inverse function, change the value of x and y of each point. The graph obtained will be a reflection across the line \(y = x\). 
 

A function must be bijective in order to have an inverse. This means it must be:

• One-to-one (injectable): Every input has a unique output.
• Onto (surjective): Every output is matched with at least one input.

Let’s look at how the bijection can be represented in a graph.

This function is not one-to-one because, as observed from its graph, two different input values result in the same output. Therefore, the function is not inverse unless its domain is restricted.
 

Finding the inverse of a function requires a clear understanding of the steps to follow.

• Replace f(x) with y in the function 

• Solve the equation for x in terms of y

• To find the inverse relationship, reciprocate x and y.

• Use the inverse notation f⁻¹(x) in place of y.
   

Example: Find the inverse of f(x) = 4x + 8.

Solution:
Given, \(f(x) = 4x + 8\). Here, we substitute y in place of f(x).

\(y = 4x + 8\\ \implies y - 8 = 4x\\ \implies x = {{(y -8)\over 4}} \)

Reciprocate x and y: 
\(\implies y = {(x - 8) \over 4}\\ \implies f^{-1} = {(x - 8)\over 4}\)

To find the inverse of a function, interchange the x and y values of each point, that is, \((x, y)\) with \((y, x)\). This creates a reflection of the original function across the line \(y = x\).

To check if two graphs are inverses, check whether they are symmetric to the line \(y = x\). This is because for every point \((x, y)\) on one graph, the point \((y, x)\) will be on its inverse.

The inverses of a few common functions are shown in the table below. This is useful to find the inverses of more complex functions. Any input values that are restricted for the inverse functions are listed in the “Corner Cases” column.
 

The inverse function can be classified into different types based on the type of original function. We will now take a closer look at each of them.

Inverse Trigonometric Functions

Inverse trigonometric functions help us find the angle whose sine, cosine, or tangent is a given value. Their domains (allowed inputs) and ranges (potential outputs) are given in the table below:

Exponential and Logarithmic Functions

A logarithmic function is the reverse of an exponential function. For example, if \(f(x) = a^x\), then its inverse is \(logₐx\). That is, they cancel each other out.

Inverse Hyperbolic Functions

Like trigonometric functions, hyperbolic functions also have inverses. The inverse hyperbolic functions are the inverses of hyperbolic functions such as sinh x, cosh x, and tanh x and their inverses are \(sinh⁻¹x, cosh⁻¹x,{\text { and }}tanh⁻¹x\).
 

Inverse functions can feel tricky and confusing at first because they involve reversing processes, but with the right approach, they become simple to understand. These tips and tricks will help students to identify, find, and verify inverse functions.

• One of the easiest methods to find inverse is switching the roles of x and y in the equation, then solve for y.
  
  
   
• After finding the inverse, verify it by checking if \( f(f^{ −1} (x))=x \) and \(f^{-1}(f(x))=x\). If both are true, your answer is correct.
  
  
   
• Use the horizontal line test to check if the function has an inverse or not. A function has an inverse only if it passes the horizontal line test, that is no horizontal line should cross the graph more than once.
  
  
   
• The inverse of a function is its mirror image across the line \(𝑦 = 𝑥\). Plot both functions to clearly see their reflection
  
  
   
• Apply inverse functions to real-world problems like converting temperatures, decoding messages, or finding time from speed and distance. Practice helps you connect theory with use.

Inverse functions have many real-world uses because they help us reverse processes, decode data, and convert one quantity into another. Here are some practical ways they are applied in different fields
 

• Inverse functions are widely used in encryption or decryption processes to keep information secure. When data is encrypted, a function transforms the original message into a coded form. To decode it, we apply the inverse of that function, which brings the message back to its original state. For example, we use the inverse of that function to decode the message or recover the original.
  
  
   
• Temperature scales such as Celsius and Fahrenheit are related through inverse functions. For example, the formula F = (9/5)C + 32 is used to convert Celsius to Fahrenheit, while its inverse C = (5/9)(F – 32) is used to convert back to Celsius.
   
• Inverse function is used in research, to reverse the logarithmic and exponential functions. For example, the decay model for radioactive decay uses exponential functions, and logarithmic (inverse) functions can be used to determine the time.
  
  
   
• Economists use inverse functions to find how price and demand affect each other. For example, if \( Q=100−2P \) shows how quantity depends on price, its inverse \(P=50−0.5Q \) tells how price changes with demand. This helps businesses set prices and predict customer behavior.
  
  
   
• In engineering, inverse functions help reverse inputs and outputs. If a machine’s speed depends on voltage, the inverse function finds the voltage needed for a certain speed. This ensures accurate control in robotics, automation, and other systems.