### Piecewise Function Analysis**Problem Statement:**Consider the following piecewise function:\[ f(x) = \begin{cases} -4 & \text{if } x \le 0 \\x^2 - 4 & \text{if } 0 < x \le 1 \\-x & \text{if } x > 1 \end{cases} \]**Questions to Answer:**a. Sketch a graph of this function.b. What are the domain and range of this function?---**Explanation:**### Graphing the Piecewise Function1. **When $ x \le 0 $:** \[ f(x) = -4 \] The value of the function is constant at $-4$ for all $x \le 0$. This creates a horizontal line at $y = -4$ extending leftwards from $x = 0$.2. **When $ 0 < x \le 1 $:** \[ f(x) = x^2 - 4 \] This describes a quadratic function, which is a parabola that opens upwards. Since this interval is from $0$ to $1$ (excluding 0 and including 1), the function will draw part of a parabola from $ x=0 $ to $ x=1 $. - At $x = 0$, $f(x) = 0^2 - 4 = -4$ (excluding this point). - At $x = 1$, $f(x) = 1^2 - 4 = -3$ (this point is included).3. **When $ x > 1 $:** \[ f(x) = -x \] This is a linear function with a negative slope. It extends indefinitely to the right, starting from $x = 1$. - At $x = 1$, $f(x) = -1$.### Domain and Range- **Domain:** The function is defined for all real numbers $x$. Therefore, the domain is: \[ (-\infty, \infty) \]- **Range:** - For $x \le 0$, the range value is \(-4\

Answered: Consider the following piecewise…

Consider the following piecewise function: -4 f(x) =x - 4 0 1 Sketch a graph of this function. What are the domain and range of this function? а. b.

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ChapterP: Prerequisites: Fundamental Concepts Of Algebra

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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

### Piecewise Function Analysis

**Problem Statement:**
Consider the following piecewise function:

\[ f(x) = \begin{cases}
-4 & \text{if } x \le 0 \\
x^2 - 4 & \text{if } 0 < x \le 1 \\
-x & \text{if } x > 1
\end{cases} \]

**Questions to Answer:**
a. Sketch a graph of this function.
b. What are the domain and range of this function?

---

**Explanation:**

### Graphing the Piecewise Function

1. **When $ x \le 0 $:**
\[
f(x) = -4
\]
The value of the function is constant at $-4$ for all $x \le 0$. This creates a horizontal line at $y = -4$ extending leftwards from $x = 0$.

2. **When $ 0 < x \le 1 $:**
\[
f(x) = x^2 - 4
\]
This describes a quadratic function, which is a parabola that opens upwards. Since this interval is from $0$ to $1$ (excluding 0 and including 1), the function will draw part of a parabola from $ x=0 $ to $ x=1 $.

- At $x = 0$, $f(x) = 0^2 - 4 = -4$ (excluding this point).
- At $x = 1$, $f(x) = 1^2 - 4 = -3$ (this point is included).

3. **When $ x > 1 $:**
\[
f(x) = -x
\]
This is a linear function with a negative slope. It extends indefinitely to the right, starting from $x = 1$.

- At $x = 1$, $f(x) = -1$.

### Domain and Range

- **Domain:** The function is defined for all real numbers $x$. Therefore, the domain is:
\[
(-\infty, \infty)
\]

- **Range:**
- For $x \le 0$, the range value is \(-4\

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