Answer
To graph the function \( f(x) \), plot a solid line from \( (2,1) \) to just below \( (6,5) \), a closed dot at \( (6,-3) \), and a horizontal line at \( y = -4 \) for \( x > 6 \).
Solution
Certainly! Let's graph the piecewise function:
\[
f(x)=\left\{
\begin{array}{lll}
x-1 & \text{for} & 2 \leq x < 6 \\
-3 & \text{for} & x = 6 \\
-4 & \text{for} & x > 6
\end{array}
\right.
\]
### Step-by-Step Graphing:
1. **Identify the Different Pieces of the Function:**
- **Line Segment:** \( f(x) = x - 1 \) for \( 2 \leq x < 6 \)
- **Point:** \( f(x) = -3 \) at \( x = 6 \)
- **Horizontal Line:** \( f(x) = -4 \) for \( x > 6 \)
2. **Plot Each Piece Separately:**
- **For \( 2 \leq x < 6 \), \( f(x) = x - 1 \):**
- Calculate endpoints:
- At \( x = 2 \): \( f(2) = 2 - 1 = 1 \)
- As \( x \) approaches 6: \( f(6^-) = 6 - 1 = 5 \)
- Plot a **solid line** from \( (2, 1) \) to just below \( (6, 5) \). Use an **open circle** at \( (6, 5) \) to indicate that \( x = 6 \) is not included in this interval.
- **At \( x = 6 \), \( f(x) = -3 \):**
- Plot a **solid dot** at \( (6, -3) \).
- **For \( x > 6 \), \( f(x) = -4 \):**
- Draw a **horizontal dashed line** starting just after \( x = 6 \) at \( y = -4 \).
- Optionally, use an **open circle** at \( (6, -4) \) if you want to indicate that \( x = 6 \) is not included in this interval.
3. **Combine All Pieces:**
Here's a conceptual representation of the graph:
```
y
|
5 | o
4 |-----------------
3 | *
2 |
1 |●
0 |________________________ x
2 6
```
- **●** at \( (2, 1) \): Start of the line segment.
- **o** at \( (6, 5) \): Open circle where the line segment approaches but doesn't include \( x = 6 \).
- **\*** at \( (6, -3) \): Solid dot representing \( f(6) = -3 \).
- **\--** line extending to the right from \( x = 6 \) at \( y = -4 \): Represents \( f(x) = -4 \) for \( x > 6 \).
### Detailed Description:
- **From \( x = 2 \) to \( x = 6 \):**
- The function increases linearly with a slope of 1.
- It starts at \( (2, 1) \) and approaches \( (6, 5) \) but does not include this point.
- **At \( x = 6 \):**
- There is a distinct value \( f(6) = -3 \), represented by a closed dot.
- **For \( x > 6 \):**
- The function is constant at \( y = -4 \).
### Summary:
- **Line Segment:** Solid line from \( (2,1) \) to \( (6,5) \) (open at \( x = 6 \)).
- **Point at \( x = 6 \):** Closed dot at \( (6,-3) \).
- **Horizontal Line for \( x > 6 \):** Dashed or solid line at \( y = -4 \).
If you have graphing software or graph paper, you can plot these points and lines accordingly to visualize the function.
Reviewed and approved by the UpStudy tutoring team
Explain
Simplify this solution
