University Calculus: Early Transcendentals (3rd Edition)

$y=x^{1/3}$ is one-to-one.
*Condition for One-to-one Function (the Horizontal Line Test): The graph of function $y=f(x)$ intersects each horizontal line at most once. Here, from the graph of $y=x^{1/3}$, we can see that each horizontal line would intersect the graph only at most one time. In other words, for any value of x in the domain of $y=x^{1/3}$, there is only one corresponding value of y. Therefore, $y=x^{1/3}$ is one-to-one.