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!

---

### 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.

Explore the detailed step-by-step analysis of the function f(x) = x² for x ≥ -1. Understand its domain, range, behavior, and graph characteristics. Suitable for students and enthusiasts interested in quadratic functions and function analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation