
Graph n(x)=-1-|x|
 on the Cartesian plane to the right.

 n(x) is increasing on the interval(s):  
  

 
 n(x) is decreasing on the interval(s):  
  

 n(x)has a maximum of 
 at x = 

 
 n(x) has a minimum of 
 at x = 

The domain of n(x)
 is:  
  

The range of n (x)
 is:  
  

If any of the above do not exist, enter DNE in the answer box.

Let's break down the given function and analyze its behavior.

The function is \( n(x) = -1 - |x| \), where \( |x| \) represents the absolute value of \(x\). This absolute value function is piecewise-defined, and the expression \( -1 - |x| \) shifts the graph downward by 1 unit.

### 1. **Graph of \( n(x) = -1 - |x| \)**
The graph of this function is a "V" shape, inverted due to the negative sign, with its vertex at \( x = 0 \), since that's where the absolute value function reaches its minimum. The function decreases as you move away from 0 in both directions.

### 2. **Intervals of Increase/Decrease**
- \( n(x) \) is decreasing everywhere except at \( x = 0 \), because the absolute value makes both left and right parts of the graph slope downward.
- There are no intervals where \( n(x) \) is increasing.

Thus:
- **Increasing interval(s):** DNE
- **Decreasing interval(s):** \( (-\infty, \infty) \)

### 3. **Maximum and Minimum Values**
- Since \( n(x) \) is always decreasing as you move away from the vertex, the **maximum** value occurs at \( x = 0 \), where \( n(0) = -1 - |0| = -1 \).
- The function has no minimum because it continues to decrease indefinitely as \( |x| \) increases.

Thus:
- **Maximum value:** -1 at \( x = 0 \)
- **Minimum value:** DNE

### 4. **Domain**
The function is defined for all real numbers, so the domain is:
- **Domain:** \( (-\infty, \infty) \)

### 5. **Range**
Since the function continues decreasing indefinitely, the range starts at the maximum value and extends downward:
- **Range:** \( (-\infty, -1] \)

### Final Summary:

1. **Increasing interval(s):** DNE  
2. **Decreasing interval(s):** \( (-\infty, \infty) \)  
3. **Maximum value:** -1 at \( x = 0 \)  
4. **Minimum value:** DNE  
5. **Domain:** \( (-\infty, \infty) \)  
6. **Range:** \( (-\infty, -1] \)

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Would you like more details or have any questions?  
Here are 5 related questions:
1. How would the graph change if \( n(x) = -1 + |x| \)?
2. What happens to the maximum and minimum if \( n(x) = -2 - |x| \)?
3. Can you find the slope of \( n(x) = -1 - |x| \) for \( x > 0 \) and \( x < 0 \)?
4. How does the absolute value function affect the behavior of \( n(x) \)?
5. What would the range be if the function were shifted upward?

**Tip:** To determine the domain and range of an absolute value function, look at how the graph shifts and the extent to which it covers the vertical axis.

Graph n(x) = -1 - |x| on the Cartesian plane to the right.
n(x) is increasing on the interval(s):
n(x) is decreasing on the interval(s):
n(x) has a maximum of at x =
n(x) has a minimum of at x =
The domain of n(x) is:
The range of n(x) is:
If any of the above do not exist, enter DNE in the answer box.

Learn how to graph the function n(x) = -1 - |x| and determine its domain, range, increasing and decreasing intervals, as well as its maximum and minimum values. This problem explores absolute value functions and their graphing behavior, suitable for students in Grades 9-11.

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