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.