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Let's work through the problem step by step, starting with the exponential function \( g(x) = 2^x - 3 \).

### 1. **Graphing the Exponential Function**:
   The function \( g(x) = 2^x - 3 \) represents an exponential function with a vertical shift. Here’s what we need to do:

   - **Parent Function**: The parent function is \( f(x) = 2^x \). This is a standard exponential curve.
   - **Shift**: The term \(-3\) means the graph is shifted down by 3 units. So, instead of approaching \( y = 0 \) as \( x \to -\infty \), it approaches \( y = -3 \).

#### Key steps:
1. **Asymptote**: The horizontal asymptote is at \( y = -3 \).
2. **Two Points**: Let's find two points by choosing convenient \( x \)-values:
   
   - For \( x = 0 \): 
     \[
     g(0) = 2^0 - 3 = 1 - 3 = -2
     \]
     So, one point is \( (0, -2) \).
   
   - For \( x = 1 \): 
     \[
     g(1) = 2^1 - 3 = 2 - 3 = -1
     \]
     So, another point is \( (1, -1) \).

### 2. **Asymptote**:
   The asymptote is a horizontal line that the graph approaches but never touches. For this function, it is at \( y = -3 \).

### 3. **Domain and Range**:
   - **Domain**: The domain of any exponential function is all real numbers, so \( (-\infty, \infty) \).
   - **Range**: Since the function has been shifted down by 3 units and never reaches or goes below the asymptote, the range is \( (-3, \infty) \).

### Summary:
- **Two points on the graph**: \( (0, -2) \) and \( (1, -1) \)
- **Asymptote**: \( y = -3 \)
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (-3, \infty) \)

Would you like more details on any part of this, or a graph to visualize it?

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Here are five questions to expand your understanding:
1. What happens to the graph if we change the base of the exponential function?
2. How does adding or subtracting a constant outside the exponential function affect its graph?
3. What is the effect of multiplying the function by a constant, such as \( g(x) = 3 \cdot 2^x - 3 \)?
4. Can you describe the behavior of the function as \( x \to -\infty \) and \( x \to \infty \)?
5. How would the graph change if the shift was upwards, say \( g(x) = 2^x + 2 \)?

### Tip:
When graphing exponential functions, always start by identifying the asymptote and the general shape of the parent function. Then apply any shifts or stretches accordingly.

Graph the exponential function g(x) = 2^x - 3. To do this, plot two points on the graph of the function, and also draw the asymptote. Additionally, give the domain and range of the function using interval notation.

Learn how to graph the exponential function g(x) = 2^x - 3, including plotting two points, drawing the horizontal asymptote, and determining the domain and range in interval notation. Perfect for high school students studying exponential functions.

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