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!