Answer :
To graph the given function
[tex]\[ f(x) = \begin{cases} x + 11 & \text{for } x < -5 \\ -x + 8 & \text{for } x > -1 \end{cases} \][/tex]
follow these steps:
### Step 1: Identify the Break Points
The function changes its definition at [tex]\( x = -5 \)[/tex] and [tex]\( x = -1 \)[/tex]. This means we need to consider the behavior of the function to the left of [tex]\( -5 \)[/tex] and to the right of [tex]\( -1 \)[/tex].
### Step 2: Graph [tex]\( f(x) = x + 11 \)[/tex] for [tex]\( x < -5 \)[/tex]
For the interval [tex]\( x < -5 \)[/tex]:
1. Choose a few values of [tex]\( x \)[/tex] in the interval [tex]\( x < -5 \)[/tex] and compute [tex]\( f(x) \)[/tex].
2. For example, choose [tex]\( x = -6, -7, -8 \)[/tex]:
- [tex]\( f(-6) = -6 + 11 = 5 \)[/tex]
- [tex]\( f(-7) = -7 + 11 = 4 \)[/tex]
- [tex]\( f(-8) = -8 + 11 = 3 \)[/tex]
Now, plot these points on a Cartesian plane:
- [tex]\( (-6, 5) \)[/tex]
- [tex]\( (-7, 4) \)[/tex]
- [tex]\( (-8, 3) \)[/tex]
Draw a straight line passing through these points extending to the left.
### Step 3: Graph [tex]\( f(x) = -x + 8 \)[/tex] for [tex]\( x > -1 \)[/tex]
For the interval [tex]\( x > -1 \)[/tex]:
1. Choose a few values of [tex]\( x \)[/tex] in the interval [tex]\( x > -1 \)[/tex] and compute [tex]\( f(x) \)[/tex].
2. For example, choose [tex]\( x = 0, 1, 2 \)[/tex]:
- [tex]\( f(0) = -0 + 8 = 8 \)[/tex]
- [tex]\( f(1) = -1 + 8 = 7 \)[/tex]
- [tex]\( f(2) = -2 + 8 = 6 \)[/tex]
Now, plot these points on a Cartesian plane:
- [tex]\( (0, 8) \)[/tex]
- [tex]\( (1, 7) \)[/tex]
- [tex]\( (2, 6) \)[/tex]
Draw a straight line passing through these points extending to the right.
### Step 4: Consider the Function's Domain and Range
The function has gaps between [tex]\( -5 \)[/tex] and [tex]\( -1 \)[/tex] where it is not defined. Ensure that there are no lines, points or connecting curves in this interval.
### Step 5: Final Graph
Plot the graphs of the two functions:
- A line [tex]\( f(x) = x + 11 \)[/tex] for [tex]\( x < -5 \)[/tex]
- A line [tex]\( f(x) = -x + 8 \)[/tex] for [tex]\( x > -1 \)[/tex]
Make sure to label the axes and provide a clear indication of the function definitions:
```markdown
|
10+
| .......(x-intercept for -x+8 at x=8)
| /
f(x) = 8 - x| /
| /
| /
5 + | /
| /
| /
-------------------(-5)---------
|
|
|
```
Summarize and cross-check the points for the graph with defined intervals to ensure consistency. These include the domains [tex]\( x < -5 \)[/tex] and [tex]\( x > -1 \)[/tex] with no defined values in between.
[tex]\[ f(x) = \begin{cases} x + 11 & \text{for } x < -5 \\ -x + 8 & \text{for } x > -1 \end{cases} \][/tex]
follow these steps:
### Step 1: Identify the Break Points
The function changes its definition at [tex]\( x = -5 \)[/tex] and [tex]\( x = -1 \)[/tex]. This means we need to consider the behavior of the function to the left of [tex]\( -5 \)[/tex] and to the right of [tex]\( -1 \)[/tex].
### Step 2: Graph [tex]\( f(x) = x + 11 \)[/tex] for [tex]\( x < -5 \)[/tex]
For the interval [tex]\( x < -5 \)[/tex]:
1. Choose a few values of [tex]\( x \)[/tex] in the interval [tex]\( x < -5 \)[/tex] and compute [tex]\( f(x) \)[/tex].
2. For example, choose [tex]\( x = -6, -7, -8 \)[/tex]:
- [tex]\( f(-6) = -6 + 11 = 5 \)[/tex]
- [tex]\( f(-7) = -7 + 11 = 4 \)[/tex]
- [tex]\( f(-8) = -8 + 11 = 3 \)[/tex]
Now, plot these points on a Cartesian plane:
- [tex]\( (-6, 5) \)[/tex]
- [tex]\( (-7, 4) \)[/tex]
- [tex]\( (-8, 3) \)[/tex]
Draw a straight line passing through these points extending to the left.
### Step 3: Graph [tex]\( f(x) = -x + 8 \)[/tex] for [tex]\( x > -1 \)[/tex]
For the interval [tex]\( x > -1 \)[/tex]:
1. Choose a few values of [tex]\( x \)[/tex] in the interval [tex]\( x > -1 \)[/tex] and compute [tex]\( f(x) \)[/tex].
2. For example, choose [tex]\( x = 0, 1, 2 \)[/tex]:
- [tex]\( f(0) = -0 + 8 = 8 \)[/tex]
- [tex]\( f(1) = -1 + 8 = 7 \)[/tex]
- [tex]\( f(2) = -2 + 8 = 6 \)[/tex]
Now, plot these points on a Cartesian plane:
- [tex]\( (0, 8) \)[/tex]
- [tex]\( (1, 7) \)[/tex]
- [tex]\( (2, 6) \)[/tex]
Draw a straight line passing through these points extending to the right.
### Step 4: Consider the Function's Domain and Range
The function has gaps between [tex]\( -5 \)[/tex] and [tex]\( -1 \)[/tex] where it is not defined. Ensure that there are no lines, points or connecting curves in this interval.
### Step 5: Final Graph
Plot the graphs of the two functions:
- A line [tex]\( f(x) = x + 11 \)[/tex] for [tex]\( x < -5 \)[/tex]
- A line [tex]\( f(x) = -x + 8 \)[/tex] for [tex]\( x > -1 \)[/tex]
Make sure to label the axes and provide a clear indication of the function definitions:
```markdown
|
10+
| .......(x-intercept for -x+8 at x=8)
| /
f(x) = 8 - x| /
| /
| /
5 + | /
| /
| /
-------------------(-5)---------
|
|
|
```
Summarize and cross-check the points for the graph with defined intervals to ensure consistency. These include the domains [tex]\( x < -5 \)[/tex] and [tex]\( x > -1 \)[/tex] with no defined values in between.
