Inverse Functions | Definition, Condition for Inverse and Examples

Inverse Functions are an important concept in mathematics. An inverse function basically reverses the effect of the original function. If you apply a function to a number and then apply its inverse, you get back the original number. For example, if a function turns 2 into 5, the inverse function will turn 5 back into 2.

In mathematical terms:

If functions f(x) and g(x) are inverses of each other, then f(x) = y only if g(y) = x. 

g(f(x)) = x

The figure given below describes a function and its inverse:

The above function is represented as f(x) and takes some input values and gives an output. The inverse of this function is denoted by f^-1(x) takes the output values produced by f(x) and converts them back to the input values. For example, let's say f(x) = 2x. It doubles the number which is given as input, its inverse should make them half to get back the input. f^-1(x) = x/2.

Inverse Function Example

Let's say we have a function f(x) = x². Now we are asked to find out the inverse of this function. This function is squaring its inputs, we know we need to take the square root for calculating the inverse.

f(x) =  x²
f^-1(x) = √x
f^-1(f(x))  = √x² = ±x

We see that there are two answers possible, which one to choose? In such cases, the inverse is not possible. So, there are things we need to notice for the functions for which inverses are possible. Also, the function whose inverse exists is called an invertible function.

Conditions for Inverse Function

For a function to have an inverse, the necessary and sufficient condition is

Function must be Bijective(One-One and Onto).

For example, let's check the following graph for bijection.

This function has same values at two different values of input, this means function is not one-one. Thus, we won't be able to find it's inverse without restricting its domain.

Also Read: What is an inverse function?

How to Find the Inverse of a Function?

To find the inverse of a function, we need to follow the following steps:

Step 1: Substitue f(x) in the given function by "y".

Step 2: Solve for "x" for the newly formed equation.

Step 3: Switch the positions of "x" and "y".

Step 4: Substitute the y with notation of inverse function f ^-1(x).

Example: Find the inverse of f(x) = 6x + 10. 

Solution: 

We know, f(x) = 6x + 10. Let's substitute y in place of f(x). 

y = 6x + 10
⇒ y - 10 = 6x
⇒ x = (y - 10)/6
⇒ y = (x - 10)/6
⇒ f ^-1(x) = (x - 10)/6 

Inverses of Common Functions 

The table given below describes the inverses of some common functions which may come in handy while calculating the inverses for complex functions.

The following table represents the function, its inverse, and its corner cases where corner cases describe the values that are not allowed as input to the inverse of the function.

Inverse Functions Graphs

To understand the graph of the inverse function, let's say we have f(x) = e^x and assume it has inverse i.e., g(x). We know that the inverse of an exponential function is a logarithmic function. So, g(x) = log_ex. The figure below shows the graph for both of the functions. 

We can see that both graphs are mirror images of each other with respect to the line y = x.

Note: Inverse of a function is a mirror image of the function when seen through the line y = x. There is no shortcut way to plot the graph of the inverse function if the graph of the original function is not given.

Also Read: Graph of inverse trigonometry system

Inverse Function Types

There are various types of inverse functions for common functions, some of these types are discussed as follows:

Inverse Trigonometric Function

Inverse Trigonometric Functions are the inverse functions of the trigonometric ratios, and the table for the range and domain of all the Inverse Trigonometric Functions is as follows:

Exponential and Logarithm Function

Another example of inverse pair is the exponential and logarithm function, both are inverse of each other. For an exponential function f(x) = a^x, its inverse is given by logarithm i.e., log_ax, and vice versa.

Inverse Hyperbolic Function

Similar to the Inverse Trigonometric Function, there are inverse hyperbolic functions, which are the inverse of the hyperbolic trigonometric function i.e., sinh x, cosh x, tanh x, and so on. Inverse Hyperbolic Function are sinh^-1, cosh^-1x, tanh^-1x, cosech^-1x, coth^-1x, and sech^-1x.

Solved Examples of problems on Inverse Functions

Problem 1: Find the inverse of the function f(x) = \frac{x + 4}{2x + 1}

Solution: 

 f(x) = \frac{x + 4}{2x + 1}

Substituting f(x) with y

y = \frac{x + 4}{2x + 1}

⇒ y(2x + 1) = x + 4
⇒ 2xy + y = x + 4 
⇒ x(2y - 1) = 4 - y

⇒ x = \frac{4 - y}{2y - 1}

Thus, f^-1(y) = \frac{4 - y}{2y - 1}

Problem 2: Find the inverse of the function f(x) = ln x + 5. 

Solution:  

f(x) = lnx + 5 

Substituting the f(x) with y
y = lnx + 5 
⇒ lnx= y - 5
⇒ x = e^{(y - 5)}
 f^-1(y) = e^{(y - 5)}

Problem 3: Find the inverse of the following function and draw its graph: f(x) = e^x + 20

Solution: 

f(x) = e^x + 20

Substituting the f(x) with y 
⇒y = e^x + 20
⇒y - 20 = e^x
⇒ln(y - 20) = x

f^-1(y) = ln(y - 20)

The figure below, shows the graphs for f(x) and it's inverse. 

Notice that y > 20 for this function. 

Problem 4: State whether the statement is True or False. For the given function f(x) = x² + 4, the inverse does not exist for all values of x. 

Solution:

We know that f(x) = x² + 4 is not bijective. For example, 

f(-2) = 8 and f(2) = 8. So, the inverse for this function cannot exist for all values of x. Thus, this statement is called False. 

Problem 5: Find the inverse for the following function: 

f(x) = \frac{x}{5x + 1}

Solution:

f(x) = \frac{x}{5x + 1}        

Substituting f(x) with y. 

y = \frac{x}{5x + 1}

⇒ y = \frac{x}{5x + 1}

⇒ y(5x + 1) = x
⇒ 5xy + y = x
⇒ x(5y - 1) = -y 

⇒ x =\frac{-y}{5y - 1}

Thus, f^-1(y) = \frac{-y}{5y - 1}

Related Read:

• Types of Functions
• Domain and Range of a Function
• Relations and Functions
• Exponential Function