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

Question

Classify the following functions as injection, surjection or bijection:
$f : R \rightarrow R,$ defined by $f(x) = x^3 + 1$

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Answer

$f : R \rightarrow R,$ defined by $f(x) = x^3 + 1$
Injection test: Let x and y be any two elements in the domain $(R),$ such that $f(x) = f(y).$
$f(x) = f(y)$
$x^3 + 1 = y^3 + 1$
$x^3 = y^3$
$x = y$
So, $f$ is 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).
$f(x) = y$
$x^3 + 1 = y$
$\text{x}=\sqrt[3]{\text{y}-1}\in\text{R}$
So, $f$ is a surjection.
So, $f$ is a bijection.

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