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

Question

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

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Answer

$f : N → N$ given by $f(x) = x^2$ Let $x_1 = x_2$ for $\text{x}_1,\text{ x}_2\in\text{N}$$\text{x}_1^2=\text{x}_2^2\Rightarrow\ \text{f}(\text{x}_1)=\text{f}(\text{x}_2)$
$\therefore f$ is one-one.
Surjectivity: Since $f$ takes only square value like $1, 4, 9, 16 .....$
So, non-perfect square values in $N (\infty-\text{domain})$ do not have pre image in domain $N.$
Thus, $f$ is not onto.

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