jmc

For $x\le 0,$ the expression $x^2$ takes all nonnegative values. Meanwhile, for $x>0,$ the expression $x^3$ takes all positive values, so $x^3-5$ takes all values in the interval $(-5,\infty ).$ It follows that the range of $f(x)$ is the union of the two intervals $[0,\infty )$ and $(-5,\infty ).$ Because the second interval contains the first one, the range of $f(x)$ is simply $\boxed{(-5,\infty )}.$