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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