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
  ⇒ y - 8 = 4x
  ⇒ x = (y - 8)/4
Reciprocate x and y: 
  ⇒ y = (x - 8)/4
  ⇒ f⁻¹(x) = (x - 8)/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) = aˣ, 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, and tanh⁻¹x.
 

Inverse functions are used in various fields. Here are a few real-life applications of inverse functions.

 

• Inverse functions are widely applied in encryption or decryption processes. For example, we use the inverse of that function to decode the message or recover the original.
  
   
• We use the inverse of a function, in temperature conversion that is from Fahrenheit to Celsius and vice versa. 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.