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

    \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{3}}}\]

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

    \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=~{{\mathbf{x}}^{\mathbf{3}}}\]

Given f: Z → Z given by 

    \[f\left( x \right)\text{ }=~{{x}^{3}}\]

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 (Z), such that f(x) = f(y)

f(x) = f(y)

    \[{{x}^{3}}~=~{{y}^{3}}\]

x = y

So, f is an injection.

Surjection condition:

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

f(x) = y

    \[{{x}^{3}}=~y\]

    \[x\text{ }=\sqrt[3]{y}\]

which may not be in Z.

For example, if y =

    \[3\]

,

    \[x\text{ }=\sqrt[3]{3}\]

is not in Z.

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

i

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