## 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:

- Algebraically
- Graphically

**Representing equation of inverse relation algebraically: **

Consider fx = x^{2}

y = x^{2}

x = y^{2} (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 = x^{2} and y = ±√x** **

The graph of the inverse of f is the reflection of the graph of y = x^{2} across the line y = x.

#### Restrict a domain to produce an inverse function

Consider the function y = x^{2}

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) = x^{2 }is y = ±√x.

If the **domain** of f(x) = x^{2} 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 ⇒ x^{2}=y−2

x^{2}+2=y

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

f−1(x)=x^{2}+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)=x^{3}+1 are inverse functions.

**Solution: **

Given

f(x)=∛x−1 and g(x)=x^{3}+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 πr^{3}

Rewrite the formula to find the length of the radius.

4/3 πr^{3} = v

πr^{3} = 3/4 v

r^{3} = 3/4π v

r = ∛3/4𝜋 V

One liter is equivalent to 1000cm^{3}.

So, 4.5L is equivalent to 4500cm^{3}.

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.

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