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Classify the following function as injection, surjection or bijection: f: N → N given by

    \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{2}}}\]

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

    \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{2}}}\]

Given f: N → N, given by

    \[~f\left( x \right)\text{ }=~{{x}^{2}}\]

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

Injection condition:

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

f(x) = f(y)

    \[{{x}^{2~}}=\text{ }{{y}^{2}}\]

x = y (We do not get 

    \[\pm \]

because x and y are in N that is natural numbers)

So, f is an injection.

Surjection condition:

Let y be any element in the co-domain (N), such that f(x) = y for some element x in N (domain).

f(x) = y

    \[{{x}^{2}}=~y\]

    \[x\text{ }=\text{ }\sqrt{y}\]

, which may not be in N.

For example, if y =

    \[3\]

,

x =

    \[\sqrt{3}\]

 is not in N.

So, f is not a surjection.

Also f is not a bijection.

i

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