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

    \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=~\mathbf{x}~-\text{ }\mathbf{5}\]

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

    \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=~\mathbf{x}~-\text{ }\mathbf{5}\]

Given f: Z → Z, defined by 

    \[f\left( x \right)\text{ }=~x~\text{ }5\]

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 (Z), such that f(x) = f(y).

f(x) = f(y)

    \[x\text{ }~5\text{ }=~y\text{ }~5\]

x = y

So, f is an injection.

Surjection test:

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

    \[x\text{ }~5\text{ }=~y\]

    \[x\text{ }=\text{ }y~+\text{ }5\]

, which is in Z.

So, f is a surjection and f is a bijection

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