Classify the following functions as injection, surjection or bije… - Vidyadip

Question

Classify the following functions as injection, surjection or bijection:
$f : Z \rightarrow Z$ given by $f(x) = x^2$

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Answer

$f: Z \rightarrow Z$ given by $f(x)=x^2$
Injection test: Let $x$ and $y$ be any two elements in the domain $(Z)$, such that $f(x)=f(y) . f(x)=f(y) x^2=y^2 x= \pm y$ So, f is not 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). $f(x)=$ $y x^2=y x= \pm \sqrt{y}$ which may not be in $Z$.
For example, if $y=3$,
$\text{x}=\pm\sqrt{3}$ is not in Z.
So, f is not a bijection.

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