Question
sketch the graph of f(x)= {-2 if x ≤ -1
x-1 if -1 < x ≤ 1
-1 if x> 1
x-1 if -1 < x ≤ 1
-1 if x> 1
Answers
GPT-4o mini
To sketch the graph of the piecewise function \( f(x) \):
1. **Define the pieces of the function**:
- For \( x \leq -1 \): \( f(x) = -2 \)
- For \( -1 < x \leq 1 \): \( f(x) = x - 1 \)
- For \( x > 1 \): \( f(x) = -1 \)
2. **Analyze each piece of the function**:
- For \( x \leq -1 \):
- The graph is a horizontal line at \( y = -2 \) up to and including \( x = -1 \). This is a closed endpoint at \( (-1, -2) \).
- For \( -1 < x \leq 1 \):
- The equation \( f(x) = x - 1 \) is a linear function with a slope of 1. At \( x = -1 \), \( f(-1) = -1 - 1 = -2 \) (not included, as the interval is open on the left), and at \( x = 1 \), \( f(1) = 1 - 1 = 0 \) (included).
- Therefore, this segment will start from the point \( (-1, -2) \) (open circle) and go to the point \( (1, 0) \) (closed circle).
- For \( x > 1 \):
- The graph is a horizontal line at \( y = -1 \) starting just after \( x = 1 \). The point \( (1, -1) \) is excluded, and it extends indefinitely to the right.
3. **Plot these details**:
- Start at the point where \( x \leq -1 \) and draw a horizontal line at \( y = -2 \) from \( -\infty \) up to \( (-1, -2) \), including that point.
- From \( x = -1 \) (open circle) to \( x = 1 \) (closed circle), draw a straight line segment that increases from \( (-1, -2) \) to \( (1, 0) \).
- At \( x = 1 \), mark the point \( (1, 0) \) and draw a horizontal line at \( y = -1 \) extending to the right.
4. **Final graph**:
- The piecewise function will have three distinct parts, transitioning smoothly from one piece to the next, with specific open and closed endpoints as noted above.
Here’s a representation of the graph:
```
y
|
0 * (1, 0)
|
| /
| /
-1|_____/________________
| /
-2|---*
| x
-1 0 1
(-1, -2) zero derivative
```
In summary, the graph includes a horizontal line at \( y = -2 \) for \( x \leq -1\), a sloping line from \( (-1, -2) \) to \( (1, 0) \) for \( -1 < x \leq 1\), and a horizontal line at \( y = -1\) for \( x > 1\), with the respective endpoints.
1. **Define the pieces of the function**:
- For \( x \leq -1 \): \( f(x) = -2 \)
- For \( -1 < x \leq 1 \): \( f(x) = x - 1 \)
- For \( x > 1 \): \( f(x) = -1 \)
2. **Analyze each piece of the function**:
- For \( x \leq -1 \):
- The graph is a horizontal line at \( y = -2 \) up to and including \( x = -1 \). This is a closed endpoint at \( (-1, -2) \).
- For \( -1 < x \leq 1 \):
- The equation \( f(x) = x - 1 \) is a linear function with a slope of 1. At \( x = -1 \), \( f(-1) = -1 - 1 = -2 \) (not included, as the interval is open on the left), and at \( x = 1 \), \( f(1) = 1 - 1 = 0 \) (included).
- Therefore, this segment will start from the point \( (-1, -2) \) (open circle) and go to the point \( (1, 0) \) (closed circle).
- For \( x > 1 \):
- The graph is a horizontal line at \( y = -1 \) starting just after \( x = 1 \). The point \( (1, -1) \) is excluded, and it extends indefinitely to the right.
3. **Plot these details**:
- Start at the point where \( x \leq -1 \) and draw a horizontal line at \( y = -2 \) from \( -\infty \) up to \( (-1, -2) \), including that point.
- From \( x = -1 \) (open circle) to \( x = 1 \) (closed circle), draw a straight line segment that increases from \( (-1, -2) \) to \( (1, 0) \).
- At \( x = 1 \), mark the point \( (1, 0) \) and draw a horizontal line at \( y = -1 \) extending to the right.
4. **Final graph**:
- The piecewise function will have three distinct parts, transitioning smoothly from one piece to the next, with specific open and closed endpoints as noted above.
Here’s a representation of the graph:
```
y
|
0 * (1, 0)
|
| /
| /
-1|_____/________________
| /
-2|---*
| x
-1 0 1
(-1, -2) zero derivative
```
In summary, the graph includes a horizontal line at \( y = -2 \) for \( x \leq -1\), a sloping line from \( (-1, -2) \) to \( (1, 0) \) for \( -1 < x \leq 1\), and a horizontal line at \( y = -1\) for \( x > 1\), with the respective endpoints.