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Classify the following function as injection, surjection or bijection: f: R → R, defined by

    \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{sin}\text{ }\mathbf{x}\]

CBSE | Class 12 | Exercise 2.1 | Function | Maths | RD Sharma
Classify the following function as injection, surjection or bijection: f: R → R, defined by

    \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{sin}\text{ }\mathbf{x}\]

Given f: R → R, defined by f(x) = sin x

Now we have to check for the given function is injection, surjection and bijection condition.

Injection test:

Let x and y be any two elements in the domain (R), such that f(x) = f(y).

f(x) = f(y)

Sin x = sin y

Here, x may not be equal to y because 

    \[sin\text{ }0\text{ }=\text{ }sin\text{ }\pi \]

.

So, 

    \[0\]

 and 

    \[\pi \]

 have the same image 

    \[0\]

.

So, f is not an injection.

Surjection test:

Range of f =

    \[[-1,1]\]

Co-domain of f = R

Both are not same.

So, f is not a surjection and f is not a bijection.

i

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