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

Question

Classify the following functions as injection, surjection or bijection : $f : N \rightarrow N$ given by $f(x) = x^3$

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Answer

$f : N \rightarrow N$ given by $f(x) = x^3$
Injection test:
Let $x$ and $y$ be any two elements in the domain $(N),$ such that $f(x) = f(y)$.
$f(x) = f(y)$
$x^3 = y^3$
$x = y$
Therefore$, f$ is an injection.
Surjection test:
Let $y$ be any element in the co$-$domain $(N),$ such that $f(x) = y$ for some element $x$ in $N\ ($domain$)$.
$f(x) = y$
$x^3 = y$
$\text{x}=\sqrt[3]{\text{y}}$ which may not be in $N$.
For example, if $y = 3,$
$\text{x}=\sqrt[3]{3}$ is not in $N$.
Therefore$, f$ is not a surjection and $f$ is not a bijection.

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