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

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

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

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

Given f: R → R, defined by 

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

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)

    \[\left| x \right|=\left| y \right|\]

    \[x\text{ }=\text{ }\pm y\]

So, f is not an injection.

Surjection test:

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

f(x) = y

    \[\left| x \right|=y\]

    \[x~=~\pm ~y~\in ~Z\]

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

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