### Instruction: Find the range of the function \[f(x) = \left\{ \begin{aligned} x^2 & \quad \text{ if } x \le 0 \\ x^3 - 5 & \quad \text{ if } x > 0. \end{aligned} \right.\].### Response: I need to find the set of all possible values of f(x) for any x in the domain of f.
The domain of f is the set of all real numbers, since there are no restrictions on x.
To find the range, I will consider the two cases: when x is negative or zero, and when x is positive.
When x is negative or zero, f(x) = x^2, which is always non-negative.
As x approaches negative infinity, x^2 also approaches positive infinity.
As x approaches zero, x^2 approaches zero.
Therefore, the range of f(x) for x <= 0 is the interval [0, infinity).
When x is positive, f(x) = x^3 - 5, which is an increasing function.
As x approaches positive infinity, f(x) also approaches positive infinity.
As x approaches zero from the right, f(x) approaches -5.
Therefore, the range of f(x) for x > 0 is the interval (-5, infinity).
Combining these two intervals, the range of f(x) for any x is the interval (-5, infinity).
This is the final answer.