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The given function is \( g(x) = 3^x - 3 \), which is an exponential function. Here's how we can approach this problem step by step:

### 1. Graphing the Function
We need to plot the graph of \( g(x) = 3^x - 3 \). Let's first calculate a few points on the graph:

- At \( x = 0 \): 
  \[
  g(0) = 3^0 - 3 = 1 - 3 = -2
  \]
  So, one point is \( (0, -2) \).

- At \( x = 1 \): 
  \[
  g(1) = 3^1 - 3 = 3 - 3 = 0
  \]
  Another point is \( (1, 0) \).

- At \( x = -1 \): 
  \[
  g(-1) = 3^{-1} - 3 = \frac{1}{3} - 3 = \frac{1}{3} - \frac{9}{3} = -\frac{8}{3}
  \]
  This gives the point \( (-1, -\frac{8}{3}) \), or approximately \( (-1, -2.67) \).

### 2. Asymptote
The horizontal asymptote of this function is \( y = -3 \), because as \( x \to -\infty \), \( 3^x \) approaches 0, so \( g(x) \) approaches \( -3 \).

### 3. Domain and Range
- **Domain**: The domain of an exponential function is all real numbers, so the domain is \( (-\infty, \infty) \).
  
- **Range**: Since the graph never reaches \( -3 \) but approaches it as \( x \to -\infty \), and goes to infinity as \( x \to \infty \), the range is \( (-3, \infty) \).

### Summary of Answers:
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (-3, \infty) \)
- **Horizontal Asymptote**: \( y = -3 \)
- Points to plot: \( (0, -2) \), \( (1, 0) \), \( (-1, -2.67) \)

Would you like further details on this or have any questions?

Here are 5 related questions to expand on this information:
1. How does changing the base of the exponential function affect its graph?
2. What happens to the graph of \( g(x) = 3^x - 3 \) if we add a constant to \( x \), like in \( g(x) = 3^{x+1} - 3 \)?
3. Can you find the y-intercept of any exponential function in general form?
4. What is the effect of a vertical shift on the range of an exponential function?
5. How can you determine the asymptote of a transformed exponential function?

**Tip**: When graphing exponential functions, always look for the horizontal asymptote first to understand how the function behaves at extreme values of \( x \).

Graph the exponential function g(x) = 3^x - 3. Plot two points on the graph, draw the asymptote, and identify the domain and range using interval notation.

Learn how to graph the exponential function g(x) = 3^x - 3, plot key points, identify the horizontal asymptote, and find the domain and range using interval notation. Suitable for high school students in grades 9-12.

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