 graph the function state its domain and range y = 3 * e^-x -2

Let's analyze and graph the function $ y = 3e^{-x} - 2 $, and determine its domain and range.

### 1. **Understanding the Function:**
   - The function is an exponential function of the form $ y = a \cdot e^{-x} + c $, where $ a = 3 $ and $ c = -2 $.
   - The base of the exponent is the natural base $ e $ (approximately 2.718), and the exponent $ -x $ indicates a reflection over the y-axis, meaning the function will decrease as $ x $ increases.
   - The function is shifted downward by 2 units due to the constant $ -2 $.

### 2. **Graphing the Function:**
   - **Asymptote:** The horizontal asymptote will be at $ y = -2 $ since the function is shifted down by 2 units.
   - **Behavior:** The graph starts high when $ x $ is negative and decreases towards the asymptote $ y = -2 $ as $ x $ increases.
   - **Key Points:**
     - At $ x = 0 $: $ y = 3e^{0} - 2 = 3(1) - 2 = 1 $
     - As $ x $ increases: $ y $ approaches $ -2 $.
     - As $ x $ decreases (large negative values of $ x $): $ y $ increases without bound.

### 3. **Domain and Range:**
   - **Domain:** The domain of the function is all real numbers, $ (-\infty, \infty) $, since the exponential function is defined for all $ x $.
   - **Range:** The range is determined by the horizontal asymptote and the direction in which the function moves. Since the graph approaches $ -2 $ but never reaches it and increases without bound as $ x $ decreases, the range is $ (-2, \infty) $.

### 4. **Plot the Graph:**
   - **x-values:** Choose a few key x-values (e.g., -2, -1, 0, 1, 2) and calculate the corresponding y-values to plot the curve.
   - **Graph Behavior:** The curve will start from above, pass through the point (0, 1), and gradually decrease towards $ y = -2 $ as $ x $ increases.

Let's visualize the graph of this function:

```python
import numpy as np
import matplotlib.pyplot as plt

# Define the function y = 3 * e^(-x) - 2
def f(x):
    return 3 * np.exp(-x) - 2

# Create an array of x values
x = np.linspace(-3, 3, 400)
y = f(x)

# Plot the function
plt.figure(figsize=(8, 6))
plt.plot(x, y, label=r"$y = 3e^{-x} - 2$", color='blue')
plt.axhline(y=-2, color='red', linestyle='--', label='Asymptote $y=-2$')
plt.axhline(0, color='black',linewidth=0.5)
plt.axvline(0, color='black',linewidth=0.5)
plt.title(r"Graph of $y = 3e^{-x} - 2$")
plt.xlabel("x")
plt.ylabel("y")
plt.ylim([-5, 5])
plt.xlim([-3, 3])
plt.legend()
plt.grid(True)
plt.show()
```

### Summary:
- **Domain:** $ (-\infty, \infty) $
- **Range:** $ (-2, \infty) $

Would you like to see a plotted graph or need further details?

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### Questions:
1. How would the graph change if the function were $ y = 3e^{x} - 2 $?
2. What is the y-intercept of the function?
3. How does the horizontal asymptote change if the constant term changes from -2 to 0?
4. How does the function behave as $ x $ approaches large positive and negative values?
5. How would the graph look if the coefficient of $ e^{-x} $ was negative?
6. How does the function change if the exponent is $ e^{-2x} $ instead of $ e^{-x} $?
7. What is the effect of multiplying the function by a constant factor?
8. How does the range of the function change if the function is $ y = 3e^{-x} + 2 $ instead of $ y = 3e^{-x} - 2 $?

**Tip:** When analyzing functions that involve exponential decay, always check for asymptotes to understand the long-term behavior of the function.

Learn how to graph and analyze the function y = 3e^(-x) - 2. Understand its domain, range, and behavior through a detailed explanation and plotted graph. Suitable for high school students.

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