# Inverse Trigonometric Functions, Objectives, is Inverse of Every Function Possible?

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## Objectives

After studying this lesson, you will be able to:

## Is Inverse of Every Function Possible?

• Take two ordered pairs of a function and

• If we invert them, we will get and

• This is not a function because the first member of the two ordered pairs is the same.

Now let us take another function:

and

Writing the inverse, we have

and

Which is a function.

Let us consider some examples from daily life.

Student Score in Mathematics

Do you think will exist?

• It may or may not be because the moment two students have the same score, will cease to be a function. Because the first element in two or more ordered pairs will be the same. So we conclude that

• Every function is not invertible.

Example:

If defined by . What will be ?

Solution:

In this case is one-to-one and onto both.

is invertible

Let

So , inverse function of i.e.,

The functions that are one-to-one and onto will be invertible.

• Let us extend this to trigonometry:

• Take . Here domain is the set of all real numbers. Range is the set of all real numbers lying between and , including and i.e. .

Consider, where (domain) and orThis is many-to-one and onto function, therefore it is not invertible.

Can be made invertible and how? Yes, if we restrict its domain in such a way that it becomes one-to-one and onto taking as

1. , or

2. , or

3. , etc.

Now consider the inverse function .

We know the domain and range of the function. We interchange domain and range for the inverse of the function. Therefore,

1. , or

2. , or

3. , etc.

Here we take the least numerical value among all the values of the real number whose sine is which is called the principle value of .

For this the only case is . Therefore, for principal value of , the domain isi.e. and range is .

Similarly, we can discuss the other inverse trigonometric functions.

 Function Domain Range(Principal value) 1 2 3 4 5 or 6 or

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