Let's go through each part of the question based on the function $f(x) = \log(x - 4) + 3$.

Part (a): Domain of $f(x)$

The function $f(x) = \log(x - 4) + 3$ is defined only when the argument of the logarithm is positive. Thus, we need: $x - 4 > 0$ Solving this inequality: $x > 4$ So, the domain of $f$ is $(4, \infty)$.

Answer for part (a): The domain is $(4, \infty)$.

Part (b): Identifying the Correct Graph

To identify the correct graph, note that:

1. The graph of $f(x) = \log(x - 4) + 3$ will have a vertical asymptote at $x = 4$, as the function is undefined there.
2. The logarithmic function increases as $x$ increases, so the curve should move upwards as $x$ moves right.
3. Adding 3 shifts the graph up by 3 units.

Based on these characteristics, option C is the correct graph, as it matches the behavior of the function with a vertical asymptote at $x = 4$ and an upward trend.

Answer for part (b): The correct graph is C.

Part (c): Range of $f(x)$

Since $f(x) = \log(x - 4) + 3$, the range of the logarithmic function $\log(x - 4)$ is $(-\infty, \infty)$. Adding 3 shifts the range of $f(x)$ to cover all real numbers.

Answer for part (c): The range is $(-\infty, \infty)$.

Would you like additional details or have any questions?

Further Questions to Explore:

1. How does the transformation $f(x) = \log(x - 4) + 3$ compare to $f(x) = \log(x)$?
2. Why does the addition of 3 affect only the range but not the domain?
3. What would happen to the graph if we subtracted 3 instead of adding it?
4. How would the domain change if the function was $f(x) = \log(x + 4) + 3$?
5. What are the characteristics of the logarithmic function that define its graph's shape?

Tip:

For logarithmic functions, shifts left or right affect the domain, while vertical shifts (up or down) alter the range.