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You are asked to graph the exponential function \( g(x) = -4^{x} - 3 \), find its asymptote, and provide the domain and range in interval notation. Let's break it down:

1. **Function Overview:**
   - The function \( g(x) = -4^x - 3 \) is a transformation of the standard exponential function.
   - It has a vertical shift downward by 3 units and a reflection due to the negative coefficient.

2. **Graphing the Function:**
   - Start with two key points on the graph. Calculate values of \( g(x) \) for simple \( x \)-values:
     - When \( x = 0 \), \( g(0) = -4^0 - 3 = -1 - 3 = -4 \).
     - When \( x = 1 \), \( g(1) = -4^1 - 3 = -4 - 3 = -7 \).
   - Plot these points: \( (0, -4) \) and \( (1, -7) \).

3. **Asymptote:**
   - The horizontal asymptote occurs where the function flattens out for large negative or positive \( x \). Since there is a vertical shift of -3, the asymptote will be \( y = -3 \).

4. **Domain:**
   - Exponential functions like this one have an unrestricted domain, meaning they are defined for all real numbers.
   - **Domain** in interval notation: \( (-\infty, \infty) \).

5. **Range:**
   - The range of \( g(x) \) is all values below the horizontal asymptote. Since the function never reaches \( y = -3 \) but approaches it from below, the range is \( (-\infty, -3) \).
   - **Range** in interval notation: \( (-\infty, -3) \).

Would you like more details on the graphing steps or clarification on any specific part of this explanation?

Here are some related questions to deepen your understanding:

1. How does the negative coefficient in \( -4^x \) affect the shape of the graph?
2. Why does the function never actually touch the asymptote \( y = -3 \)?
3. What would happen to the graph if the vertical shift were changed to a different value?
4. How would the domain and range differ for a similar function without the negative sign in front of \( 4^x \)?
5. Can you identify any key differences between \( g(x) = -4^x - 3 \) and the standard exponential function \( 4^x \)?

**Tip:** In exponential functions, always pay attention to reflections (due to negative signs) and vertical shifts to understand the graph's behavior better!

Graph the exponential function g(x) = -4^x - 3, plot two points on the graph, draw the asymptote, and provide the domain and range in interval notation.

Learn how to graph the exponential function g(x) = -4^x - 3, including finding the asymptote, domain, and range in interval notation. Suitable for high school students studying exponential functions.

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