Let's break down the tasks required based on the image.

1. Transformation of the graph of $y = e^x$ to $y = e^{x-5} + 3$:

   • Start with the base graph $y = e^x$, which is an exponential function.
   • The transformation involves two steps:
     • Horizontal shift: The $x-5$ indicates a horizontal shift to the right by 5 units.
     • Vertical shift: The $+3$ means a vertical shift upwards by 3 units.

2. Domain and Range of $y = e^{x-5} + 3$:

   • Domain: The domain of the exponential function is all real numbers $(-\infty, \infty)$, since the exponential function is defined for all real values of $x$.
   • Range: The range of the base function $y = e^x$ is $(0, \infty)$. After the vertical shift by 3, the range becomes $(3, \infty)$, because the lowest point of the graph will now be shifted from 0 to 3.

So, in interval notation:

• Domain: $(-\infty, \infty)$
• Range: $(3, \infty)$

Would you like more details on any part of this solution or further clarifications?

Here are 5 related questions to explore further:

1. How does the horizontal shift affect the asymptote of an exponential function?
2. How would the graph change if the function was $y = e^{-(x-5)} + 3$?
3. What is the behavior of the function as $x \to -\infty$ and $x \to \infty$?
4. How would you reflect the graph of $y = e^{x-5} + 3$ across the x-axis?
5. What would the domain and range be if the transformation were $y = e^{x-5} - 3$?

Tip: When transforming graphs, horizontal shifts are typically inside the function's exponent or argument, while vertical shifts are added or subtracted outside.