This problem asks you to graph the function $g(x) = 1 + e^x$, analyze its domain, range, and asymptote, and choose the correct graph among the provided options.

Steps to Solve:

1. Starting from the Graph of $e^x$: The function $g(x) = 1 + e^x$ is a vertical shift of the graph of $e^x$. The graph of $e^x$ starts from $y = 1$ when $x = 0$, approaches $y = 0$ as $x \to -\infty$, and grows exponentially as $x \to \infty$.

   The transformation $1 + e^x$ shifts the entire graph of $e^x$ up by 1 unit. So, the new graph will:

   • Start from $y = 2$ at $x = 0$ (since $e^0 = 1$, so $g(0) = 1 + 1 = 2$),
   • Approach $y = 1$ as $x \to -\infty$,
   • Grow exponentially as $x \to \infty$.

2. Domain of $g(x) = 1 + e^x$: The exponential function $e^x$ is defined for all real numbers $x$. Hence, the domain of $g(x)$ is: $\text{Domain: } (-\infty, \infty)$

3. Range of $g(x) = 1 + e^x$: Since $e^x > 0$ for all $x$, adding 1 ensures that $g(x) > 1$ for all $x$. Therefore, the range of $g(x)$ is: $\text{Range: } (1, \infty)$

4. Asymptote of $g(x) = 1 + e^x$: The exponential function $e^x$ approaches 0 as $x \to -\infty$, so the function $g(x) = 1 + e^x$ approaches $y = 1$ as $x \to -\infty$. Thus, the horizontal asymptote is: $\text{Asymptote: } y = 1$

Conclusion:

• Domain: $(-\infty, \infty)$
• Range: $(1, \infty)$
• Asymptote: $y = 1$

Now, based on these observations, the correct graph should:

• Be an upward-shifted version of $e^x$,
• Start from $y = 2$ at $x = 0$,
• Approach $y = 1$ for negative values of $x$,
• Grow rapidly as $x$ increases.

Looking at the four provided graphs:

• The second graph in the first row (right side) matches these characteristics, as it shows a rapid growth to the right, a shift up from $e^x$, and approaches a horizontal asymptote.

Would you like more details or clarification on any part? Here are 5 related questions to think about:

1. How would the graph change if the equation were $g(x) = e^x - 2$?
2. What happens to the domain and range if a negative coefficient is added to $e^x$, such as $g(x) = -e^x$?
3. How does the horizontal asymptote shift for a function like $g(x) = 3 + e^x$?
4. Can you explain why the domain of $e^x$ is always all real numbers?
5. How would you find the asymptote of a more complex exponential function like $g(x) = 2e^{x-1} + 5$?

Tip: When graphing transformed functions, always identify how shifts and stretches affect key features like intercepts, asymptotes, and growth rate.