Show that the function f:
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.