### 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 outputs of the function f, which are the values of f(x) for any x in the domain of f.
The domain of f is the entire set of real numbers, since there are no restrictions on x.
To find the range, I will consider the two cases: x ≤ 0 and x > 0.
If x ≤ 0, then f(x) = x^2, which is a parabola that opens upward and has a vertex at (0,0).
As x decreases from 0 to negative infinity, f(x) increases from 0 to positive infinity.
So the range of f(x) for x ≤ 0 is the interval [0, ∞).
If x > 0, then f(x) = x^3 - 5, which is a cubic function that is increasing and has no horizontal asymptotes.
As x increases from 0 to positive infinity, f(x) increases from -5 to positive infinity.
So the range of f(x) for x > 0 is the interval (-5, ∞).
Therefore, the range of f(x) for any x is the union of these two intervals, which is (-5, ∞).