**– $ \space sin^{- 1}$**

**– $ \space cos^{- 1}$**

**– $ \space tan^{- 1}$**

The **main objective** of this question is to find the **domain** and** range** for the **given functions**.

This question **uses** the **concept** of **range** and **domain** of **functions**. The **set among** all **values within** which a **function** is defined is **known** as its **domain**, and its **range** is the set of **all possible values**.

## Expert Answer

In this **question**, we have to find the **domain** and **range** for the **given functions**.

a) **Given that**:

\[ \space sin^{ – 1 } \]

We have to **find** the **range** and **domain** of this **function**. We know that the **set among** all **values** **within** which a **function** is defined is known as its **domain**, and its **range** is the set of all **possible values**.

**Thus**, the** domain** of $ sin^{ – 1} $ is:

\[ \space = \left[ \space – \space\frac{ \pi}{ 2 }, \space \frac{ \pi}{ 2 } \right] \]

**And** the **range** of $ sin^{ – 1 } $ is:

\[ \space = \space [- \space 1, \space 1] \]

b)**Given that**:

\[ \space cos^{ – 1 } \]

We have to **find** the **range** and **domain** of this **function**. We know that the **set among** all **values** **within** which a **function** is defined is known as its **domain**, and its **range** is the set of all **possible values**.

**Thus**, the** domain** of $ cos^{ – 1} $ is:

\[ \space = \space – \space 0, \space \pi \]

**And** the **range** of $ cos^{ – 1} $ is:

\[ \space = \space [- \space 1, \space 1] \]

c) **Given that**:

\[ \space tan^{ – 1 } \]

We have to **find** the **range** and **domain** of this **function**. We know that the **set among** all **values** **within** which a **function** is defined is known as its **domain**, and its **range** is the set of all **possible values**.

**Thus**, the** domain** of $ tan^{ – 1} $ is:

\[ \space = \left[ \space – \space\frac{ \pi}{2}, \space \frac{ \pi}{ 2 } \right] \]

**And** the **range** of $ tan^{ – 1} $ is:

\[ \space = \space [ R ]\]

## Numerical Answer

The **domain** and **range** of $ sin^{-1} $ is:

\[ \space = \space [ – \space 1, \space 1 ] ,\space\left[ \space – \space\frac{ \pi}{2}, \space \frac{ \pi}{ 2 } \right] \]

The **domain** and **range** of $cos^{-1} $ is:

\[ \space = \space [ – \space 1, \space 1 ]\space [ – \space 0, \space \pi ] \]

The **domain** and **range** of $ tan^{-1} $ is:

\[ \space = \space R \space , \space\left[ \space – \space\frac{ \pi}{2}, \space \frac{ \pi}{ 2 } \right] \]

## Example

**Find** the **range** and **domain** for the **given function**.

\[ \space = \space \frac{ 6 }{x \space – \space 4} \]

We have to **find** the **range** and **domain** for the given **function**.

**Thus**, the **range** for the** given function** is all real **numbers** without **zero,** while the **domain** for the **given function** is **all numbers** that are real **except** the **number** which is equal to $ 4 $.