The function \( f(x) = x^2 \) is defined for \( x \geq -1 \). Since the function is a simple quadratic function with a domain restriction, we can evaluate and discuss its properties:

1. **Domain:** The function is defined for all \( x \) such that \( x \geq -1 \).

2. **Range:** Since \( f(x) = x^2 \) and the smallest value \( x \) can take is \(-1\), the smallest value of \( f(x) \) is \( (-1)^2 = 1 \). Therefore, the range of the function is \( [1, \infty) \).

3. **Behavior:** The function is increasing for \( x > 0 \) and decreasing for \( x < 0 \) in the domain \( x \geq -1 \).

4. **Vertex:** The vertex of the parabola is at \( x = 0 \), where \( f(0) = 0^2 = 0 \), but this point is not within the domain of the restricted function, so it does not impact the defined range.

5. **Graph:** The graph is a portion of the parabola \( y = x^2 \), starting at the point \( (-1, 1) \) and extending indefinitely to the right.

If you want details, have any questions, or need a graphical representation, let me know!

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Here are five related questions to consider:
1. What is the derivative of \( f(x) = x^2 \) for \( x \geq -1 \)?
2. How would the function change if the domain were \( x \leq -1 \) instead?
3. What is the inverse of the function \( f(x) = x^2 \) over its defined domain?
4. How do you find the area under the curve \( f(x) = x^2 \) from \( x = -1 \) to \( x = 2 \)?
5. What are the roots of the function if the domain were unrestricted?

**Tip:** For functions with restricted domains, always check how the restriction affects the range and other properties.

Explore the quadratic function f(x) = x² defined for x ≥ -1. Learn about its domain, range, behavior, and graphical representation. Suitable for high school students, this page provides insights into quadratic functions and their properties.

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