The problem involves transforming the graph of $y = e^x$ to the graph of $y = e^{x+1} + 2$, and determining the domain and range of this transformed function.

1. Transforming the Graph:

The transformation from $y = e^x$ to $y = e^{x+1} + 2$ involves two steps:

• Horizontal shift: The expression $e^{x+1}$ represents a horizontal shift of the graph of $y = e^x$ to the left by 1 unit. This is because adding 1 inside the exponent causes the graph to move in the opposite direction of the sign.

• Vertical shift: The addition of 2 outside the exponential function, $e^{x+1} + 2$, causes the entire graph to shift upward by 2 units.

Thus, the graph of $y = e^{x+1} + 2$ is the graph of $y = e^x$ shifted 1 unit left and 2 units up.

2. Domain and Range:

• Domain: The domain of the exponential function $y = e^x$ is all real numbers. The transformation does not affect this, so the domain of $y = e^{x+1} + 2$ remains the same.

  • Domain: $(-\infty, \infty)$

• Range: The range of $y = e^x$ is $(0, \infty)$, as $e^x$ is always positive. When we apply the transformation $y = e^{x+1} + 2$, we shift the entire graph up by 2 units, so the range becomes $(2, \infty)$.

  • Range: $(2, \infty)$

Summary:

• Domain: $(-\infty, \infty)$
• Range: $(2, \infty)$

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

Here are some related questions for further understanding:

1. How does adding or subtracting values inside the exponent affect the graph of exponential functions?
2. What happens to the domain and range of exponential functions when we apply different transformations?
3. How can we identify horizontal and vertical shifts from the equation of an exponential function?
4. How do the transformations of $y = e^x$ compare with those of other types of functions, such as quadratic functions?
5. What are the key properties of exponential functions that remain unchanged under transformations?

Tip: When analyzing transformations of exponential functions, remember that shifting horizontally corresponds to modifying the exponent, while shifting vertically is achieved by adding or subtracting constants outside the function.