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.

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.