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:

pythonCopy code
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.