In this lesson, you will find all the important concepts related to inverse functions. The inverse function is a function that can be reversed from output to the original value. Here, we will discuss inverse functions, the preparation of inverse functions, properties of the inverse functions, inverse function graphs, how to find inverse functions, types of inverse functions and inverse functions using algebra.

Let us start with defining the inverse function.

Inverse function

An inverse function is a function that can be reversed. If a function y can be written with respect to x and x can be written with respect to y then it is called an inverse function. 

In such a function, say:

For example,

f(x) = 2x+3 = y

This function will also work as

x = (y – 3)/2

In this case, the inverse function will be x = f-1(y) 

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

In other words, an inverse function can be reversed into another function. If an inverse function takes you from x to y, it can also lead you from y to x.

Example:

f(3) = 2×3 +4 = 10

f-1 (10) = (10-4)/2 = 3

Inverse function graph

To draw an inverse function on a graph, you have to draw a line  y = x. Two lines will be drawn which show an inverse function and are symmetric. One line is denoted as f(x) and the other is denoted as g(x). The two lines of the inverse curve are symmetric on the line y = x.

Example:

Take an inverse function, let’s say

f(x) = 2x – 3 = y

Now, we have to find the inverse function of this function that is,

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

The function is an inverse function. To check it, put 4 as x’s value.

f(4) = 2*4 – 3 = 5 = y

You will get the value of y. Now substitute the value of y.

 f-1 (5) = (5+3)/2 = 4 = x

And you will get the value of x!

Steps for Inverse Function

As we know, a function whose output can be used to find the original value is an inverse function. These are the steps to find the inverse of a function.

• The first step is to replace f(x) with y to make the steps easier.  
• The second step is to solve the equation and find the value of y. Follow the equation rules carefully to avoid mistakes in this step.
• We have both functions now f (x) and  f-1 (x).
• Now it’s time to verify the irreversibility. Put a value of x in f(x) and find the equation with y.
• Now put the value of y and you will see that you will get the value of x.

Example,

y = 2x – 4=f(x)

Now put y in the place of f(x)

y = 2x – 4

Now find the inverse function

x = (y + 4 )/ 2

⇒ f-1  (x) = (Y + 4) / 2

Now verify the irreversibility of the function

Put 2 for the value of x

f (2) = 2 * 2 – 4 = 0 = y

Now put y’s value in inverse

f-1  (0) = (0 + 4) / 2 

⇒ 2 = x

So, these are the steps by which you can find inverse function.

Types of Inverse function

There are mainly four types of inverse functions.

1. Trigonometric functions
2. Rational functions
3. Hyperbolic functions
4. Log functions.  

Inverse functions are given below.

Inverse Trigonometric Functions

The inverse trigonometric functions are known as arc functions. The length of the arc they produce is needed to obtain the particular value. The six inverse trigonometric functions are arccosine (cos-1), arcsine (sin-1), arctangent (tan-1), arccosecant (cosec-1), arcsecant (sec-1) and arccotangent (cot-1).

Inverse Rational Function

A rational function is written in this form f(x) = P(x)/Q(x), Q(x) ≠ 0.

Inverse Hyperbolic Functions

The inverse hyperbolic function is the inverse of the hyperbolic function. The 6 inverse hyperbolic functions are sinh-1, tanh-1, csch-1, coth-1, cosh-1 and sech-1.

Inverse Logarithmic Functions and Inverse Exponential Function

The inverse of logarithmic functions are exponential functions.

For example,

f(x) =logb(x)

f-1(y) = by  (Inverse)

Inverse function using Algebra

As we know, a function whose output can be used to find the original value of the input is an inverse function. The steps to find the inverse of a function includes replacing f(x) with y and then solving the equation to find the value of y, which is the inverse function, f-1 (x).

Example

f(x) = 4 x + 8

Let f(x) = y

y = 4 x + 8

y – 8 = 4 x

x =( y – 8) / 4

f-1   (x) = (y – 8) / 4

To check the inverse function, put any value of x

Suppose x = 4

f(4) = 4 * 4 + 8 = 24 = y

f-1  (24) =( 24 – 8) / 4 = 4 = x

That’s how we find inverse function.

Conclusion

Inverse function is a function that can be reversed. Simply, if a function f with respect to x equals to y or f(x)=y, then f-1(y)=x is true; where f-1(y) is the inverse of f(x). You can get the original value from the output of the inverse function. The two lines of the inverse curve are symmetric on the x = y line. There are mainly four types of inverse functions- Trigonometric functions, Rational functions, Hyperbolic functions and log functions.

To find an inverse function put y in the place of f (x) and then find the y value and it would be an inverse function.