To find and analyze the function $f(x) = x^2$ for $x \geq -1$ step by step, we will go through the following process:

1. Define the Function and Domain:

The function is given as: $f(x) = x^2$ with the domain restriction: $x \geq -1$

This means the function is only defined for $x$ values that are greater than or equal to $-1$.

2. Evaluate the Function at the Boundary:

To understand the behavior of the function, first, evaluate it at the boundary of the domain: $f(-1) = (-1)^2 = 1$

3. Behavior of the Function:

Since $f(x) = x^2$ is a quadratic function, it is a parabola that opens upwards.

• For $x > 0$, the function increases as $x$ increases.
• For $-1 \leq x < 0$, the function decreases as $x$ approaches $0$.

However, since the domain is restricted to $x \geq -1$, we only consider the portion of the parabola starting from $x = -1$.

4. Determine the Range:

Given the domain restriction, the smallest value of $x$ is $-1$, and the smallest value of $f(x)$ is $f(-1) = 1$. As $x$ increases, $f(x)$ increases without bound.

Thus, the range of the function is: $\text{Range of } f(x) = [1, \infty)$

5. Graph the Function:

The graph of $f(x) = x^2$ for $x \geq -1$ is a parabolic curve starting at the point $(-1, 1)$ and extending infinitely to the right.

6. Summary of Key Properties:

• Domain: $x \geq -1$
• Range: $[1, \infty)$
• Behavior: Decreasing for $-1 \leq x < 0$, increasing for $x \geq 0$.

If you would like a graph or further analysis, feel free to ask!

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Related Questions:

1. How do you find the derivative of $f(x) = x^2$ for $x \geq -1$?
2. What happens to the graph if the domain is changed to $x > 0$?
3. How do you find the minimum value of $f(x) = x^2$ over its domain?
4. How does restricting the domain affect the inverse of the function?
5. What is the integral of $f(x) = x^2$ from $x = -1$ to $x = 1$?

Tip: When analyzing functions with restricted domains, always check the behavior at the endpoints and within the interval.