Answer
(a) "... one-to-one function."
(b) $g^{-1}(x)=x^\frac{1}{3}$
Work Step by Step
(a) Due to the definition of an inverse function, for a function to have an inverse function it should be one-to-one function (Also explained previously in chapter 2.8).
So we have answer "one-to-one function".
For a one-to-one function to exist each $y$ value should be connected to only one $x$ value (For arbitrary $y$ there should be only one $x$).
$f(x)=x^2$
in this case if $x=-2$ or $x=2$, we get the same $y$ value. (One $y$ value gets two different $x$ value, so it is not one-to-one function). Which means that it has no inverse function.
$g(x)=x^3$
In this case there is no two $x$ value which gives us the same $y$ value, so its one-to-one function, and so it has an inverse function.
For a better visualization see the image below taken from the textbook chapter 2.8 (Note, there is $f(x)$ instead of $g(x)$ and vice versa).
(b) To calculate the inverse function of $g(x)=x^3$, we will first write it in terms of $y$ and $x$, then change their positions and solve for $y$:
$y=x^3$
$x=y^3$
$y=\sqrt[3] x$
$y=x^\frac{1}{3}$
$g^{-1}(x)=x^\frac{1}{3}$
