By definition:

To find x-intercept of a graph , let y be zero and solve the equation for x.

$$\begin{align*} y^{2}&= x^{3}-4\cdot x\\ 0^{2}&=x^{3}-4\cdot x \\ 0&=x^{3}-4\cdot x \\ 0&=x\cdot \left(x^{2} -4\right) \\ 0&=x\cdot \left(x +2\right) \cdot \left(x-2 \right) \\ \end{align*}$$

Then, x-intercept are $x=0$ ,$x=-2$ ,$x= 2$.

And to find y-intercept of a graph , let x be zero and solve the equation for y.

$$\begin{align*} y^{2}&= x^{3}-4\cdot x\\ y^{2}&= 0^{3}-4\cdot 0 \\ y^{2}&=0 \\ y&=0\\ \end{align*}$$