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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.

Graph the function g(x) = 1 + e^x by starting from the graph of y = e^x.

Learn how to graph the exponential function g(x) = 1 + e^x by starting from the graph of y = e^x. This problem involves analyzing the domain, range, and horizontal asymptote of the function. Suitable for high school students studying exponential functions.

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