Answer :
To graph the function [tex]\( g \)[/tex], let's analyze it in two parts based on the given piecewise definition.
The function [tex]\( g \)[/tex] is defined as:
- [tex]\( g(x) = -x \)[/tex] if [tex]\( x \neq 2 \)[/tex]
- [tex]\( g(x) = -4 \)[/tex] if [tex]\( x = 2 \)[/tex]
### Step-by-Step Solution:
1. Plotting [tex]\( g(x) = -x \)[/tex] for [tex]\( x \neq 2 \)[/tex]:
- For this part, you will plot the line [tex]\( y = -x \)[/tex] for all [tex]\( x \)[/tex] except at [tex]\( x = 2 \)[/tex].
- This line has a slope of [tex]\(-1\)[/tex], indicating that for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 1.
2. Plot the special point [tex]\( g(2) = -4 \)[/tex]:
- At [tex]\( x = 2 \)[/tex], instead of the point on the line [tex]\( y = -x \)[/tex] (which would be [tex]\( -2 \)[/tex]), we place a single point at [tex]\( (2, -4) \)[/tex].
### Detailed Steps:
#### 1. Plot [tex]\( g(x) = -x \)[/tex]:
- Draw a line through the origin with a slope of [tex]\(-1\)[/tex]. This line will have points such as:
- [tex]\( (0, 0) \)[/tex]
- [tex]\( (1, -1) \)[/tex]
- [tex]\( (-1, 1) \)[/tex]
- [tex]\( (3, -3) \)[/tex]
- [tex]\( (-3, 3) \)[/tex]
- and so forth.
Since the equation [tex]\( y = -x \)[/tex] is valid for all [tex]\( x \)[/tex] except at [tex]\( x = 2 \)[/tex], we can draw the line continuously, but we must leave a gap at [tex]\( x = 2 \)[/tex].
#### 2. Plot the special point:
- At [tex]\( x = 2 \)[/tex], plot the point [tex]\( (2, -4) \)[/tex]. This point will not be on the line [tex]\( y = -x \)[/tex] but instead at the specified value [tex]\( -4 \)[/tex] by the piecewise definition. Represent this as a dot on the graph located at [tex]\( (2, -4) \)[/tex].
#### 3. Assemble the Graph:
- Draw the line [tex]\( y = -x \)[/tex] across the entire range but exclude the point [tex]\( x = 2 \)[/tex].
- Place an open circle (to indicate that the point is excluded from [tex]\( y = -x \)[/tex]) at [tex]\( (2, -2) \)[/tex] and then plot a solid dot at [tex]\( (2, -4) \)[/tex].
### Final Graph:
- You will visually have a straight line [tex]\( y = -x \)[/tex] with a gap at [tex]\( x = 2 \)[/tex].
- The open circle at [tex]\( (2, -2) \)[/tex] indicates that the function does not take this value.
- A single solid point at [tex]\( (2, -4) \)[/tex] represents the value of the function at [tex]\( x = 2 \)[/tex].
### Graph Example:
Here's what the graph approximately looks like:
```
y
↑
8 |
6 |
4 |
2 | [tex]\( \circ (2, -2) \)[/tex]
0 |--------------------(-x)--------------
-2| * (2, -4)
-4|
-6|
-8|
----------------------------------→ x
-10 -8 -6 -4 -2 0 2 4 6 8 10
```
This graph captures all the information about the function [tex]\( g \)[/tex], displaying the line [tex]\( y = -x \)[/tex] with an exception at [tex]\( x=2 \)[/tex] where the function value is instead [tex]\( -4 \)[/tex].
The function [tex]\( g \)[/tex] is defined as:
- [tex]\( g(x) = -x \)[/tex] if [tex]\( x \neq 2 \)[/tex]
- [tex]\( g(x) = -4 \)[/tex] if [tex]\( x = 2 \)[/tex]
### Step-by-Step Solution:
1. Plotting [tex]\( g(x) = -x \)[/tex] for [tex]\( x \neq 2 \)[/tex]:
- For this part, you will plot the line [tex]\( y = -x \)[/tex] for all [tex]\( x \)[/tex] except at [tex]\( x = 2 \)[/tex].
- This line has a slope of [tex]\(-1\)[/tex], indicating that for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 1.
2. Plot the special point [tex]\( g(2) = -4 \)[/tex]:
- At [tex]\( x = 2 \)[/tex], instead of the point on the line [tex]\( y = -x \)[/tex] (which would be [tex]\( -2 \)[/tex]), we place a single point at [tex]\( (2, -4) \)[/tex].
### Detailed Steps:
#### 1. Plot [tex]\( g(x) = -x \)[/tex]:
- Draw a line through the origin with a slope of [tex]\(-1\)[/tex]. This line will have points such as:
- [tex]\( (0, 0) \)[/tex]
- [tex]\( (1, -1) \)[/tex]
- [tex]\( (-1, 1) \)[/tex]
- [tex]\( (3, -3) \)[/tex]
- [tex]\( (-3, 3) \)[/tex]
- and so forth.
Since the equation [tex]\( y = -x \)[/tex] is valid for all [tex]\( x \)[/tex] except at [tex]\( x = 2 \)[/tex], we can draw the line continuously, but we must leave a gap at [tex]\( x = 2 \)[/tex].
#### 2. Plot the special point:
- At [tex]\( x = 2 \)[/tex], plot the point [tex]\( (2, -4) \)[/tex]. This point will not be on the line [tex]\( y = -x \)[/tex] but instead at the specified value [tex]\( -4 \)[/tex] by the piecewise definition. Represent this as a dot on the graph located at [tex]\( (2, -4) \)[/tex].
#### 3. Assemble the Graph:
- Draw the line [tex]\( y = -x \)[/tex] across the entire range but exclude the point [tex]\( x = 2 \)[/tex].
- Place an open circle (to indicate that the point is excluded from [tex]\( y = -x \)[/tex]) at [tex]\( (2, -2) \)[/tex] and then plot a solid dot at [tex]\( (2, -4) \)[/tex].
### Final Graph:
- You will visually have a straight line [tex]\( y = -x \)[/tex] with a gap at [tex]\( x = 2 \)[/tex].
- The open circle at [tex]\( (2, -2) \)[/tex] indicates that the function does not take this value.
- A single solid point at [tex]\( (2, -4) \)[/tex] represents the value of the function at [tex]\( x = 2 \)[/tex].
### Graph Example:
Here's what the graph approximately looks like:
```
y
↑
8 |
6 |
4 |
2 | [tex]\( \circ (2, -2) \)[/tex]
0 |--------------------(-x)--------------
-2| * (2, -4)
-4|
-6|
-8|
----------------------------------→ x
-10 -8 -6 -4 -2 0 2 4 6 8 10
```
This graph captures all the information about the function [tex]\( g \)[/tex], displaying the line [tex]\( y = -x \)[/tex] with an exception at [tex]\( x=2 \)[/tex] where the function value is instead [tex]\( -4 \)[/tex].
