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# Inverse function

Sep 15, 2022

## 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.

### 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.

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