Let’s solve the given problem step by step.

1. Transforming $y = e^x$ into $y = -e^x - 5$:

• Start with the graph of $y = e^x$:

  • This is the basic exponential graph, where it grows exponentially as $x$ increases and approaches 0 as $x \to -\infty$.

• Apply the following transformations:

  1. Reflection over the x-axis: Multiply by $-1$, resulting in $y = -e^x$. The graph flips vertically.
  2. Vertical shift downward by 5 units: Subtract 5, giving $y = -e^x - 5$. This moves the entire graph 5 units downward.

2. Domain of $y = -e^x - 5$:

The domain of the exponential function $e^x$ is all real numbers, and transformations (reflection and vertical shifts) do not affect the domain.
Domain: $(-\infty, \infty)$.

3. Range of $y = -e^x - 5$:

The range of $e^x$ is $(0, \infty)$. After reflection over the x-axis, the range of $-e^x$ becomes $(-\infty, 0)$. Subtracting 5 shifts it downward by 5 units. Thus, the range of $y = -e^x - 5$ is:
Range: $(-\infty, -5)$.

Final Answer:

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

Do you want a detailed explanation of how the transformations affect the graph?
Here are five related questions you might find helpful:

1. How do you graph $y = e^x$ and apply transformations step-by-step?
2. What happens to the domain and range when reflecting exponential graphs?
3. How does a vertical shift affect the graph of a function?
4. What are the key points and asymptotes of $y = -e^x - 5$?
5. Can this process be generalized for other exponential transformations?

Tip: When transforming graphs, always track how the key features (like intercepts and asymptotes) shift with each transformation.