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