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The given problem asks to graph the exponential function \( g(x) = 4^x + 2 \), plot two points, draw the asymptote, and provide the domain and range of the function in interval notation.

### Step-by-Step Solution:

1. **Exponential Function:** 
   The given function is \( g(x) = 4^x + 2 \). This is an exponential function where the base is 4 and there is a vertical shift upwards by 2.

2. **Plotting Two Points:**
   To plot the graph, we need two points. Let's calculate these by substituting different values of \( x \) into the function:
   
   - For \( x = 0 \):
     \[
     g(0) = 4^0 + 2 = 1 + 2 = 3
     \]
     So, the point is \( (0, 3) \).
   
   - For \( x = 1 \):
     \[
     g(1) = 4^1 + 2 = 4 + 2 = 6
     \]
     So, the point is \( (1, 6) \).

3. **Asymptote:**
   The function \( g(x) = 4^x + 2 \) has a horizontal asymptote. Since the graph of \( 4^x \) approaches zero as \( x \to -\infty \), the horizontal asymptote is shifted up by 2 units. Thus, the horizontal asymptote is:
   \[
   y = 2
   \]

4. **Domain:**
   The exponential function is defined for all real values of \( x \), so the domain is:
   \[
   (-\infty, \infty)
   \]

5. **Range:**
   Since \( 4^x \) is always positive and we add 2 to it, the smallest value of the function occurs when \( x \to -\infty \), where \( 4^x \) approaches zero. Therefore, the range is:
   \[
   (2, \infty)
   \]

### Summary:
- Points: \( (0, 3) \), \( (1, 6) \)
- Asymptote: \( y = 2 \)
- Domain: \( (-\infty, \infty) \)
- Range: \( (2, \infty) \)

Would you like more details on graphing or any of the steps?

### Follow-Up Questions:
1. What happens to the graph as \( x \to -\infty \)?
2. How does the base of the exponential function affect the graph's growth rate?
3. Can you explain the effect of the constant "2" in the function \( 4^x + 2 \)?
4. How would the graph change if the function was \( g(x) = 4^x - 2 \)?
5. What is the general form of an exponential function?

**Tip:** Always identify the asymptote in exponential functions to better understand their long-term behavior.

Graph the exponential function g(x) = 4^x + 2. To do this, plot two points on the graph of the function, and also draw the asymptote. Then, click on the graph-a-function button. Additionally, give the domain and range of the function using interval notation.

Learn how to graph the exponential function g(x) = 4^x + 2. This problem covers plotting points, identifying the horizontal asymptote, and finding the domain and range in interval notation. Suitable for students in Grades 9-12.

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