You are asked to graph the exponential function $g(x) = 2^{x-3}$, plot two points on the graph, identify the asymptote, and determine the domain and range using interval notation.

Steps to solve:

1. Equation of the function:
   The function is $g(x) = 2^{x-3}$. This is an exponential function where the base is 2, and the exponent is $x - 3$. The graph is a transformation of the basic exponential function $2^x$.

2. Plotting two points:
   To find two points, we can substitute values of $x$ into the function.

   • For $x = 3$: $g(3) = 2^{3-3} = 2^0 = 1$ So, one point is $(3, 1)$.

   • For $x = 4$: $g(4) = 2^{4-3} = 2^1 = 2$ Another point is $(4, 2)$.

   These two points are $(3, 1)$ and $(4, 2)$.

3. Asymptote:
   The asymptote of an exponential function $a^{x-h}$ is horizontal. Since there is no vertical shift in the given function, the asymptote will be at: $y = 0$ This is because, as $x \to -\infty$, the function $g(x) = 2^{x-3} \to 0$.

4. Domain:
   The domain of an exponential function is always all real numbers, as there are no restrictions on $x$: $\text{Domain: } (-\infty, \infty)$

5. Range:
   The range of this function is all positive values since an exponential function like $2^{x-3}$ never touches or crosses the $x$-axis but approaches 0: $\text{Range: } (0, \infty)$

Would you like me to create a graph for this function or go into further details?

Here are 5 related questions for further exploration:

1. What transformations occur to shift the function $g(x) = 2^x$ to $g(x) = 2^{x-3}$?
2. How does changing the base of the exponential function affect its graph?
3. How can you determine the asymptote of an exponential function with a vertical shift?
4. What are the key differences between exponential growth and exponential decay functions?
5. How would the graph of $g(x) = 2^{x-3}$ change if a negative coefficient was added, e.g., $g(x) = -2^{x-3}$?

Tip: When graphing exponential functions, always start by identifying key points like the y-intercept and transformations for easy graphing.