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Find the domain and range of the following functions.

– $ \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 $.

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