Let’s solve this step by step.

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Part 1: Graph $g(x) = 3^x - 6$ and Asymptote

1. The base function is $f(x) = 3^x$. Transform this function by subtracting 6:

   • The graph of $f(x) = 3^x - 6$ shifts the entire graph of $f(x) = 3^x$ downward by 6 units.

2. Equation of the Asymptote:

   • The horizontal asymptote of $f(x) = 3^x$ is $y = 0$.
   • After the vertical shift downward by 6, the new horizontal asymptote becomes $y = -6$.

3. Domain and Range:

   • The domain of $f(x) = 3^x - 6$ is all real numbers: $(-\infty, \infty)$.
   • The range of $f(x) = 3^x - 6$ is $(-6, \infty)$, because the smallest value $f(x)$ approaches is just above $-6$ (never touches $-6$).

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Answers to the Subparts:

Part 2: Equation of the Asymptote

The equation of the asymptote is: $y = -6$

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Part 3: Domain

The domain of $g(x) = 3^x - 6$ is: $(-\infty, \infty)$

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Part 4: Range

The range of $g(x) = 3^x - 6$ is: $(-6, \infty)$

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Graph

I will now graph $g(x) = 3^x - 6$ along with its asymptote.

import numpy as np import matplotlib.pyplot as plt

Define the function and x-values

x = np.linspace(-4, 4, 500) g_x = 3**x - 6

Plot the function

plt.figure(figsize=(8, 6)) plt.plot(x, g_x, label="g(x) = $3^x - 6$", linewidth=2) plt.axhline(y=-6, color='r', linestyle='--', label="Asymptote: y = -6")

Set graph details

plt.title("Graph of g(x) = $3^x - 6$", fontsize=14) plt.xlabel("x", fontsize=12) plt.ylabel("g(x)", fontsize=12) plt.axhline(0, color='black', linewidth=0.5, linestyle='--') plt.axvline(0, color='black', linewidth=0.5, linestyle='--') plt.ylim(-10, 10) plt.legend(fontsize=12) plt.grid(alpha=0.3)

Display the plot

plt.show() <<ImageDisplayed>>successHere is the graph of $g(x) = 3^x - 6$, along with the horizontal asymptote $y = -6$ (dashed line).

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Summary

1. Asymptote Equation: $y = -6$
2. Domain: $(-\infty, \infty)$
3. Range: $(-6, \infty)$

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Would you like additional details or have any questions?

Here are 5 related questions:

1. How do we identify horizontal asymptotes for other exponential functions?
2. What happens if we shift the graph $3^x - 6$ upwards or downwards further?
3. Can we find the x-intercept of $g(x) = 3^x - 6$?
4. What if we replace the base of 3 with another number (e.g., $2^x$ or $5^x$)?
5. How would the graph change with a horizontal shift (e.g., $g(x) = 3^{x+2} - 6$)?

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Tip: Subtracting or adding constants to a function shifts its graph vertically, while multiplying/dividing scales it vertically or horizontally.