Identify the domain and then graph the function.
f left parenthesis x right parenthesis equals RootIndex 3 StartRoot x EndRoot minus 2 


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.

Identify the domain and then graph the function. f(x) = ∛x - 2

Explore the domain and graph of the function f(x) = ∛x - 2. Understand the properties of cube root functions and how transformations affect their graphs. This guide covers essential algebraic concepts and is suitable for students in Grades 10-12.

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