Is the set $f=\{(x^3,x):x\in\mathbb R\}$ a function from $\mathbb R$ to $\mathbb R$? I need to find out if $f$ is a function or not. That is, whether or not the first coordinate of the ordered pair occurs only once in $f$. If yes, then it is a function.
My answer is yes, $f$ is a function because $f=x^{1/3}$ has a different value for every $x$.
Is this enough or do I need more explanation?

Let $y = g(x):= x^3$,  $x \in \mathbb{R}$.
Domain$_g = \mathbb{R}; $ Range$_g = \mathbb{R}$.
$g$ is injective and surjective, I.e. bijective. 
Injective: $g$ is strictly monotonic.
Surjective: $g$ is continuous on $[a,b]$ , 
for any $a \in \mathbb{R}.$
$\Rightarrow :$
An inverse function $g^{-1}$ exists, continuous and strictly monotonic.
$g^{-1}(y) = y^{1/3},$ $y \in \mathbb{R}$.
Any similarity to your function $f?$
