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.