Question

Show that the function $f:R\to R$ given by $f\left(x\right)=x^3$ is injective

Answer 1

A function is a one-to-one function if and only if each second element corresponds to one and only one first element. (each x and y value is used only once)
Use the horizontal line test to determine if a function is a one-to-one function.
If ANY horizontal line intersects your original function in ONLY ONE location, your function will be a one-to-one function and its inverse will also be a function.

Also, let $x,y\in R$ such that

$f\left(x\right)=f\left(y\right)$

$\Rightarrow x^3=y^3$

$\Rightarrow x=y$

Hence, $f$ is injective.