Study the function below and then answer the questions that follow. \[ f(x)=\left\{\begin{array}{ll}-x-2, & x

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Study the function below and then answer the questions that follow. \[ f(x)=\left\{\begin{array}{ll}-x-2, & x<-2 \ -x^{2}, & -2

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To analyze the function $$ f(x) $$, we will examine its features, including its domain, range, continuity, and any critical points.

### Step 1: Identify the Domain
The function $$ f(x) $$ is defined piecewise for three intervals:
1. $$ x < -2 $$
2. $$ -2 < x < 0 $$
3. $$ x \geq 0 $$

Thus, the domain of $$ f(x) $$ is all real numbers, $$ (-\infty, \infty) $$.

### Step 2: Identify the Range
We will evaluate the output of $$ f(x) $$ in each interval:

1. **For $$ x < -2 $$**:
   $$
   f(x) = -x - 2
   $$
   As $$ x $$ approaches $$-2$$ from the left, $$ f(x) $$ approaches $$ -(-2) - 2 = 0 $$. As $$ x $$ goes to $$-\infty$$, $$ f(x) $$ goes to $$ \infty $$. Therefore, the range for this part is $$ (0, \infty) $$.

2. **For $$ -2 < x < 0 $$**:
   $$
   f(x) = -x^2
   $$
   This is a downward-opening parabola. The maximum value occurs at $$ x = 0 $$ where $$ f(0) = 0 $$ and the minimum occurs as $$ x $$ approaches $$-2$$ where $$ f(-2) = -(-2)^2 = -4 $$. Thus, the range for this part is $$ (-4, 0) $$.

3. **For $$ x \geq 0 $$**:
   $$
   f(x) = x
   $$
   This is a linear function that starts at $$ f(0) = 0 $$ and increases without bound. Therefore, the range for this part is $$ [0, \infty) $$.

### Step 3: Combine the Ranges
Combining the ranges from all three intervals:
- From $$ x < -2 $$: $$ (0, \infty) $$
- From $$ -2 < x < 0 $$: $$ (-4, 0) $$
- From $$ x \geq 0 $$: $$ [0, \infty) $$

The overall range of $$ f(x) $$ is:
$$
(-4, \infty)
$$

### Step 4: Check for Continuity
To check for continuity at the boundaries $$ x = -2 $$ and $$ x = 0 $$:

1. **At $$ x = -2 $$**:
   - Left limit: $$ \lim_{x \to -2^-} f(x) = -(-2) - 2 = 0 $$
   - Right limit: $$ \lim_{x \to -2^+} f(x) = -(-2)^2 = -4 $$
   - Since the left limit does not equal the right limit, $$ f(x) $$ is not continuous at $$ x = -2 $$.

2. **At $$ x = 0 $$**:
   - Left limit: $$ \lim_{x \to 0^-} f(x) = -0^2 = 0 $$
   - Right limit: $$ \lim_{x \to 0^+} f(x) = 0 $$
   - Since both limits are equal, $$ f(x) $$ is continuous at $$ x = 0 $$.

### Step 5: Identify Critical Points
To find critical points, we need to find where the derivative $$ f'(x) $$ is zero or undefined.

1. **For $$ x < -2 $$**:
   $$
   f'(x) = -1
   $$
   (constant, no critical points)

2. **For $$ -2 < x < 0 $$**:
   $$
   f'(x) = -2x
   $$
   Setting $$ f'(x) = 0 $$:
   $$
   -2x = 0 \implies x = 0
   $$
   (but $$ x = 0 $$ is not in this interval)

3. **For $$ x \geq 0 $$**:
   $$
   f'(x) = 1
   $$
   (constant, no critical points)

### Summary of Features
- **Domain**: $$ (-\infty, \infty) $$
- **Range**: $$ (-4, \infty) $$
- **Continuity**: Not continuous at $$ x = -2 $$, continuous at $$ x = 0 $$
- **Critical Points**: None in the intervals, but $$ x = 0 $$ is a point of interest.

If you have any specific questions or need further analysis, feel free to ask!
The function $$ f(x) $$ is defined as: $$ f(x) = \begin{cases} -x - 2 & \text{if } x < -2 \ -x^2 & \text{if } -2 < x < 0 \ x & \text{if } x \geq 0 \end{cases} $$ **Domain**: All real numbers, $$ (-\infty, \infty) $$ **Range**: $$ (-4, \infty) $$ **Continuity**: Discontinuous at $$ x = -2 $$, continuous at $$ x = 0 $$ **Critical Points**: None in the defined intervals

Study the function below and then answer the questions that follow. \[ f(x)=\left\{\begin{array}{ll}-x-2, & x<-2 \\ -x^{2}, & -2

Ask by Bird Estrada. in the United States

Feb 28,2025

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Study the function below and then answer the questions that follow. \[ f(x)=\left\{\begin{array}{ll}-x-2, & x<-2 \\ -x^{2}, & -2 Ask by Bird Estrada. in the United States Feb 28,2025