To solve this problem, let’s analyze the given equation and determine the domain and range of the function $y = e^{x+3} + 5$.

Step 1: Understanding the transformations

1. The base graph is $y = e^x$.
2. The equation $y = e^{x+3} + 5$ involves two transformations:
   • Horizontal shift: Adding $+3$ in the exponent shifts the graph to the left by 3 units.
   • Vertical shift: Adding $+5$ outside shifts the graph up by 5 units.

Step 2: Domain

The domain of $e^x$ is all real numbers because the exponential function is defined for all $x$. Shifting it horizontally does not affect the domain. Therefore: $\text{Domain: } (-\infty, \infty)$

Step 3: Range

For the basic function $y = e^x$, the range is $(0, \infty)$ because exponential functions are always positive. After the vertical shift up by 5 units, the range of $y = e^{x+3} + 5$ becomes: $\text{Range: } (5, \infty)$

Final Answer:

• Domain: $(-\infty, \infty)$
• Range: $(5, \infty)$

Let me know if you'd like a further breakdown or additional visualizations!

1. What is the effect of a horizontal shift on a graph?
2. How does a vertical shift affect the range of a function?
3. Why is the domain of an exponential function always all real numbers?
4. What happens to the asymptote of the graph when you shift it vertically?
5. How would the graph of $y = e^{x+3} + 5$ differ from $y = e^{x+3} - 2$?

Tip: Always check how transformations affect the asymptotes of a graph to identify the range!