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The problem provided asks for an analysis of the function \( f(x) = e^x + 3 \). Specifically, it involves:

1. Graphing the function using transformations starting from the base function \( y = e^x \).
2. Finding the domain, range, and horizontal asymptote.

### Step 1: Transformations
The base function is \( y = e^x \), which is an exponential function with the following key properties:
- It has a horizontal asymptote at \( y = 0 \).
- Its domain is \( (-\infty, \infty) \) and its range is \( (0, \infty) \).

The function given, \( f(x) = e^x + 3 \), can be understood as a vertical shift of the base function. This shift moves the entire graph upwards by 3 units. Therefore:
- The horizontal asymptote, originally at \( y = 0 \), is now at \( y = 3 \).

### Step 2: Domain
The domain of the function \( e^x + 3 \) is determined by the fact that the exponential function \( e^x \) is defined for all real numbers. Hence, the domain of \( f(x) \) is:
\[
(-\infty, \infty)
\]

### Step 3: Range
The range of \( f(x) = e^x + 3 \) is obtained by observing that \( e^x \) is always positive (i.e., \( e^x > 0 \)). Therefore, adding 3 results in:
\[
f(x) > 3
\]
Thus, the range of the function is:
\[
(3, \infty)
\]

### Step 4: Horizontal Asymptote
The horizontal asymptote, as discussed earlier, is:
\[
y = 3
\]

### Final Answer
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (3, \infty) \)
- **Horizontal Asymptote**: \( y = 3 \)

Would you like more details on graphing the function or any other aspect? Here are some follow-up questions to deepen understanding:

1. How do we graph transformations of exponential functions in general?
2. What effect do horizontal shifts have on the domain and range of \( e^x \)?
3. What happens to the horizontal asymptote if the transformation is more complex (e.g., \( f(x) = e^{x-2} + 5 \))?
4. Can the range of an exponential function ever include negative values?
5. How does the base of an exponential function (e.g., \( 2^x \) instead of \( e^x \)) affect its transformations?

**Tip**: Always keep in mind that vertical shifts only affect the range and asymptotes, while the domain of exponential functions remains unaffected by such transformations.

Begin with the graph of y = e^x. Use transformations to graph the function f(x) = e^x + 3. Determine its domain, range, and horizontal asymptote.

Learn how to graph the exponential function f(x) = e^x + 3, including its domain, range, and horizontal asymptote. This problem covers key concepts like exponential functions and vertical shifts, ideal for students in Grades 10-12.

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