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We are asked to graph the function \( g(x) = 2 + \log_4(x + 3) \), plot two points, draw the asymptote, and find the domain and range using interval notation. Let's break it down step by step.

### Step 1: Understanding the function
The function \( g(x) = 2 + \log_4(x + 3) \) is a logarithmic function with a vertical shift of 2 units upwards and a horizontal shift due to \( x + 3 \).

### Step 2: Domain
For \( \log_4(x + 3) \) to be defined, the argument of the logarithm must be greater than 0:
\[
x + 3 > 0 \implies x > -3
\]
Thus, the domain of the function is \( (-3, \infty) \).

### Step 3: Asymptote
Logarithmic functions have a vertical asymptote where the argument inside the logarithm equals zero. Setting \( x + 3 = 0 \), we get:
\[
x = -3
\]
So, the vertical asymptote is at \( x = -3 \).

### Step 4: Range
Since the logarithmic function can take all real values (as \( \log_4(x+3) \) can range from \( -\infty \) to \( \infty \) and we have a vertical shift of 2), the range is:
\[
(-\infty, \infty)
\]

### Step 5: Points on the graph
We need to calculate the values of \( g(x) \) at two points. Let's choose \( x = -2 \) and \( x = 1 \):

- At \( x = -2 \):
\[
g(-2) = 2 + \log_4(-2 + 3) = 2 + \log_4(1) = 2 + 0 = 2
\]
So, one point is \( (-2, 2) \).

- At \( x = 1 \):
\[
g(1) = 2 + \log_4(1 + 3) = 2 + \log_4(4) = 2 + 1 = 3
\]
So, another point is \( (1, 3) \).

### Step 6: Graphing
You can now plot the points \( (-2, 2) \) and \( (1, 3) \), draw the vertical asymptote at \( x = -3 \), and sketch the curve, which will have a logarithmic shape (increasing but flattening out as \( x \to \infty \)).

### Final Answer:
- **Domain**: \( (-3, \infty) \)
- **Range**: \( (-\infty, \infty) \)
- **Asymptote**: \( x = -3 \)
- **Two points**: \( (-2, 2) \) and \( (1, 3) \)

Do you want more details or have any questions?  
Here are 5 related questions to extend the topic:
1. How does changing the base of the logarithm affect the shape of the graph?
2. What happens to the graph if we add or subtract a constant inside the logarithmic function, like \( \log_4(x+5) \)?
3. How would the function change if the logarithm had a negative coefficient, such as \( g(x) = -\log_4(x + 3) + 2 \)?
4. What is the effect of shifting the graph horizontally or vertically on the domain and range?
5. How can we calculate the inverse of this function?

**Tip**: When graphing logarithmic functions, always start by identifying the vertical asymptote and plotting points around it to get the right shape.

Graph the logarithmic function g(x) = 2 + log_4(x + 3). To do this, plot two points on the graph of the function, and also draw the asymptote. Additionally, give the domain and range of the function using interval notation.

Learn how to graph the logarithmic function g(x) = 2 + log_4(x + 3). This solution covers key concepts like plotting points, drawing the vertical asymptote, and identifying the domain and range. Suitable for students in Grades 10-12.

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