How do I sketch the function? ### Piecewise Function and Its GraphIn this example, we are given a piecewise function $ f(x) $ defined as follows:\[f(x) =\begin{cases} -x - 6 & \text{if } x \leq -3 \\-3 & \text{if } -3 < x \leq 4 \\1.5 - 7 & \text{if } x > 4 \end{cases}\]We are required to sketch the graph of this piecewise function based on the given conditions.### Analysis of Each Piece1. **For $ x \leq -3 $:** The function is $ f(x) = -x - 6 $. - This is a linear function with a negative slope. - As $ x $ decreases, $ f(x) $ increases according to the equation. 2. **For $ -3 < x \leq 4 $:** The function is $ f(x) = -3 $. - This is a constant function, meaning for all $ x $ in this interval, $ f(x) $ is always -3. 3. **For $ x > 4 $:** The function is $ f(x) = 1.5 - 7 $. - This simplifies to $ f(x) = -5.5 $, indicating another constant function where $ f(x) $ is always -5.5 for $ x $ greater than 4.### Graph ExplanationThe provided graph paper has both horizontal and vertical axes marked with grid lines. Here's how to sketch the graph based on the piecewise function:1. **For the interval $ x \leq -3 $:** - Plot the line $ f(x) = -x - 6 $. - For example, when $ x = -3 $, $ f(x) = -(-3) - 6 = 3 - 6 = -3 $. - Extend the line with that slope to the left of $ x = -3 $.2. **For the interval $ -3 < x \leq 4 $:** - Draw a horizontal line at $ f(x) = -3 $ - This extends from just greater than $ x = -3 $ to \( x = 4 \

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Description: How do I sketch the function? ### Piecewise Function and Its GraphIn this example, we are given a piecewise function $ f(x) $ defined as follows:\[f(x) =\begin{cases} -x - 6 & \text{if } x \leq -3 \\-3 & \text{if } -3 < x \leq 4 \\1.5 - 7 & \text{i

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Sketch a graph メ-ルーX -6 if X< -3 if -34x_L 1.5-7 if X >4 f(x)= -3

Algebra and Trigonometry (6th Edition)

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Author:Robert F. Blitzer

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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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Question

How do I sketch the function?

### Piecewise Function and Its Graph

In this example, we are given a piecewise function $ f(x) $ defined as follows:

\[
f(x) =
\begin{cases}
-x - 6 & \text{if } x \leq -3 \\
-3 & \text{if } -3 < x \leq 4 \\
1.5 - 7 & \text{if } x > 4
\end{cases}
\]

We are required to sketch the graph of this piecewise function based on the given conditions.

### Analysis of Each Piece

1. **For $ x \leq -3 $:** The function is $ f(x) = -x - 6 $.
- This is a linear function with a negative slope.
- As $ x $ decreases, $ f(x) $ increases according to the equation.

2. **For $ -3 < x \leq 4 $:** The function is $ f(x) = -3 $.
- This is a constant function, meaning for all $ x $ in this interval, $ f(x) $ is always -3.

3. **For $ x > 4 $:** The function is $ f(x) = 1.5 - 7 $.
- This simplifies to $ f(x) = -5.5 $, indicating another constant function where $ f(x) $ is always -5.5 for $ x $ greater than 4.

### Graph Explanation

The provided graph paper has both horizontal and vertical axes marked with grid lines. Here's how to sketch the graph based on the piecewise function:

1. **For the interval $ x \leq -3 $:**
- Plot the line $ f(x) = -x - 6 $.
- For example, when $ x = -3 $, $ f(x) = -(-3) - 6 = 3 - 6 = -3 $.
- Extend the line with that slope to the left of $ x = -3 $.

2. **For the interval $ -3 < x \leq 4 $:**
- Draw a horizontal line at $ f(x) = -3 $
- This extends from just greater than $ x = -3 $ to \( x = 4 \

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