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

Doorsteptutor material for IMO-Level-2 is prepared by world's top subject experts: fully solved questions with step-by-step explanation- practice your way to success.

Download PDF of This Page (Size: 199K) ↧

## 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

, or

, or

, 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,

, or

, or

, 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 |