Inverse function

Key Concepts

• The inverse of a relation.
• An equation of an inverse relation.
• The domain and range of an inverse function and restrict the domain of a function.
• An equation of an inverse function.
• Verify inverse functions by using composition.
• Use inverse function to rewrite a formula.

Domain and Range 

Domain  

The set of all possible values which qualify as inputs to a function or can also be defined as the entire set of values possible for independent variables.  

Range

The set of all the output of a function is known as the range of the function or the entire set of all values possible as outcomes of the dependent variable.  

Consider the following set of ordered pairs:  

{(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}

The domain is {1, 2, 3, 4, 5}.  

The range is {2, 4, 6, 8, 10}. 

Question 1: Find the domain and range of the functions given below:  

• √3x+1
• 5x−3
• 2|x−7|

Solution 1a:  

Given function √3x+1

3x+1≥0

x≥−1/3

So, the domain of the function √3x+1 is x≥−1/3.  

And the range of the function √3x+1 is y≥0.  

Solution 1b:  

Given function 5x−3

The domain of the function 5x−3 is the set of all real numbers. 

The range of the function 5x−3 is the set of all real numbers.  

Solution 1c:  

Given function 2|x−7|

So, the domain of the function 2|x−7| is the set of all real numbers. 

And the range of the function 2|x−7| is y≥0.  

Question 2: Fill the table as given below:  

Solution 2:  

When x=0 , then y = a = 8−3×0 =8

When y=2 , then 2 = 8−3b

3b = 8−2

b = 2

When x = 3 , then y = c = 8−3×3 = −1

Inverse of a relation

Inverse relation

An inverse relation is formed when the roles of the independent and dependent variables are reversed.  

Consider a tabular column:  

Here x is the independent variable and y is the dependent variable.  

Inverse function  

If an inverse relation of a function, f , is itself a function, it is called the inverse function of f.  

An inverse function is written as f−1(x).  

Consider a tabular column:  

Here, the domain of f do not have the same image, then the inverse of f is a function.  

An equation of an inverse relation  

Equation of inverse relation can be represented in two ways: 

1. Algebraically  
2. Graphically 

Representing equation of inverse relation algebraically:  

Consider fx = x²

y = x²

x = y² (Switch the roles of x and y)

y = ±√x (Solve for y)

The inverse of f can be represented algebraically by the equation y = ±√x.

Representing equation of inverse relation graphically

The graphs of  y = x² and y = ±√x 

The graph of the inverse of f is the reflection of the graph of y = x² across the line y = x.  

Restrict a domain to produce an inverse function

Consider the function y = x²

For a relation to be a function, no two values of x should have the same value as y.  

If a function has two x−values for the same y−value, its inverse will not be a function.  

Restrict the domain of a function such that its inverse also becomes a function. 

The inverse relation of f(x) = x² is y = ±√x. 

If the domain of f(x) = x² is restricted to x≥0, then the inverse is the function defined as f−1(x)=√x. 

Equation of an inverse function

Consider a function √𝒇𝒙=𝒙−𝟐

. 

From the graph, there is no horizontal line that intersects the graph at more than one point. 

When the graph is reflected over the line 𝒚=𝒙 to produce an inverse, there will be no vertical line that will intersect the graph at more than one point.  

So, the inverse relation will be a function.  

Let, y = x−2

Switch the roles of x and y and then solve for y. 

x=√y−2 ⇒ x²=y−2

x²+2=y

So, the inverse of f(x)=√x−2 is a function,

f−1(x)=x²+2,  x≥0.

Here the graph of f and f−1 are both functions and are reflections over the line y = x. 

Use composition to verify inverse functions

The functions f(x) and g(x) are said to be inverse functions of each other if   

(𝒇o𝒈)(𝒙) =𝒙 and  

(gof)(x) = x

Verification:  

Let f(x) be a function and its inverse gx=f−1(x) be the inverse function of f(x). 

Consider, (fog)(x)=f(g(x))

= f(f⁻¹(x))

= x

Now consider,

(gof)(x)=g(f(x))

=f⁻¹( f(x))

=x

(fog)(x)=x & (gof)(x)=x

So, the functions f(x) and g(x) are said to be inverse functions. 

Example:  

Use composition to determine whether f(x)=∛x−1 and g(x)=x³+1 are inverse functions.  

Solution:  

Given

f(x)=∛x−1 and g(x)=x³+1

To be inverse functions,

(fog)(x)=x and (gof)(x)=x. 

Rewrite Formula 

Example:  

A sculpture artist is making an ice sculpture of Earth for a display. He created a mould that can hold 4.5 L of ice. What will be the radius of the ice sculpture if he fills the mould all the way?  

Solution:  

The volume of the sphere, V = 4/3 πr³

Rewrite the formula to find the length of the radius.  

4/3 πr³ = v

πr³ = 3/4 v

r³ = 3/4π v

r = ∛3/4𝜋 V

One liter is equivalent to 1000cm³.  

So, 4.5L is equivalent to 4500cm³.  

r = ∛3/4𝜋 V

≈10.2 cm.

The ice sculpture mould will have a radius of about 10 cm.  

So, in the original equation the value of V depends on the value of r.  

Exercise

• Identify the inverse relation. Is it a function?

• Find an equation of the inverse function of fx= 2-∛x+1.
• Use composition to determine whether fx=3x+12, gx=1/3x-4 are inverse functions.

Concept Summary

What we have learned

• Represent the inverse of a relation.
• Find an equation of an inverse relation.
• Determine the domain and range of an inverse function and restrict the domain of a function.
• Find an equation of an inverse function.
• Use composition to verify inverse functions.
• Use inverse function to rewrite a formula.