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Let's break down the function \( f(x) = 4 \log_4 (x - 3) + 2 \) and find the domain, range, and equation of the asymptote.

### 1. **Domain**:
The function involves a logarithm, \( \log_4(x - 3) \), and logarithmic functions are only defined when their arguments are greater than zero. Therefore, we need:
\[
x - 3 > 0 \implies x > 3
\]
Thus, the domain of the function is \( (3, \infty) \).

### 2. **Range**:
Logarithmic functions have a range of \( (-\infty, \infty) \), and the transformations applied to the function \( 4 \log_4(x - 3) + 2 \) do not affect this. Therefore, the range of the function remains \( (-\infty, \infty) \).

### 3. **Equation of the Asymptote**:
The logarithmic function has a vertical asymptote where the argument of the logarithm equals zero. In this case, that occurs at:
\[
x - 3 = 0 \implies x = 3
\]
Thus, the vertical asymptote is \( x = 3 \).

### Conclusion:
- Domain: \( (3, \infty) \)
- Range: \( (-\infty, \infty) \)
- Vertical asymptote: \( x = 3 \)

Now, matching this with the given choices:
The correct answer is:
- **Domain: \( (3, \infty) \)**
- **Range: \( (-\infty, \infty) \)**
- **Equation of asymptote: \( x = 3 \)**

This corresponds to the **last option** in the list.

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Would you like more details on this explanation or have any specific questions?

Here are five related questions to explore further:
1. How would the domain change if the logarithmic argument was \( (x - 5) \)?
2. What happens to the range if we add a constant inside the logarithm, like \( \log_4(x - 3 + 1) \)?
3. How does the base of the logarithm (e.g., \( \log_2 \) vs. \( \log_4 \)) affect the function's shape?
4. What transformations occur to the graph when adding 2 to the entire logarithmic function?
5. How do vertical asymptotes change in other types of logarithmic or rational functions?

**Tip**: Remember that for logarithmic functions, the argument must always be greater than zero to define the domain properly.

Let f(x) = 4log4(x - 3) + 2. Find the domain, range, and equation of the asymptote.

Learn how to find the domain, range, and vertical asymptote for the function f(x) = 4log4(x - 3) + 2. This logarithmic function involves understanding transformations, domain restrictions, and vertical asymptotes. Suitable for high school students in Grades 10-12.

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