Show that the function f: R R given by f(x) = x^3 is injective. - Vidyadip

Question

Show that the function $f: R \rightarrow R$ given by $f(x) = x^3$ is injective.

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Answer

$f: R → R$ is given as $f(x) = x^3$
Suppose $f(x) = f(y),$ where $\text{x},\text{y}\in\text{R}.$
$\Rightarrow x^3 = y^3...(1)$
Now, we need to show that $x = y.$
Suppose $\text{x}\neq\text{y},$ their cubes will also not be equal.
$\Rightarrow\text{x}^3\neq\text{y}^3$
However, this will be a contradiction to $(1).$
$\therefore x = y$
Hence, $f$ is injective.

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