To identify the domain of the function $f(x) = \sqrt[3]{x} - 2$, we need to analyze the domain of the cube root function.

Step 1: Identify the Domain

For cube roots, there are no restrictions on the input $x$ because cube roots are defined for all real numbers (positive, negative, and zero). Therefore, the domain of $f(x) = \sqrt[3]{x} - 2$ is all real numbers.

Domain: $(-\infty, \infty)$

Step 2: Graph the Function

The graph of $f(x) = \sqrt[3]{x} - 2$ can be plotted as follows:

• The function is based on $\sqrt[3]{x}$, which passes through the origin (0,0).
• The transformation $-2$ means the entire graph of $\sqrt[3]{x}$ is shifted down by 2 units.
• The curve approaches negative infinity on the left as $x \to -\infty$, and increases gradually on the right as $x \to \infty$.

If you'd like, I can plot this graph for you.

Let me know if you would like the graph or more details! Here are some related questions:

1. How does the transformation $f(x) - 2$ affect the shape of a general function?
2. What is the range of $f(x) = \sqrt[3]{x} - 2$?
3. Can cube root functions have horizontal asymptotes?
4. How would the graph change if the function was $f(x) = \sqrt[3]{x} + 2$?
5. How does the domain of cube roots differ from square roots?

Tip: Cube root functions, unlike square roots, can accept negative inputs, which is why their domain is all real numbers.