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