To analyze the function $f(x) = 9x^2 \ln(2x)$, we start by examining the domain of $f(x)$.

Step 1: Determine the Domain

The function $f(x) = 9x^2 \ln(2x)$ has two parts:

1. $9x^2$, which is defined for all real values of $x$.
2. $\ln(2x)$, which is defined only when $2x > 0$, or equivalently $x > 0$.

Since $\ln(2x)$ is undefined for $x \leq 0$, we conclude that the domain of $f(x)$ is all $x > 0$.

So, the answer is: A. The domain is $(0, \infty)$.

Part 2: Sketch the Graph of $f(x)$

To sketch the graph, let’s analyze some key aspects of $f(x) = 9x^2 \ln(2x)$:

1. Domain: As determined, the domain is $(0, \infty)$.
2. Intercepts: The function $f(x)$ has no $y$-intercept because it is undefined at $x = 0$. To find the $x$-intercept, set $f(x) = 0$. This happens when $9x^2 \ln(2x) = 0$, which implies $\ln(2x) = 0$. Solving $\ln(2x) = 0$ gives $2x = 1$ or $x = \frac{1}{2}$.
   • Therefore, the $x$-intercept is $\left( \frac{1}{2}, 0 \right)$.
3. End Behavior: As $x \to \infty$, $x^2$ grows much faster than $\ln(2x)$, so $f(x) \to \infty$.
4. Derivative Analysis (for critical points):
   • To find where $f(x)$ is increasing or decreasing, take the derivative $f'(x)$ and analyze its sign. (This part involves a bit of calculus and would help identify maxima or minima.)

Using these points, you could sketch a rough graph showing:

• The curve starts from the x-intercept $\left( \frac{1}{2}, 0 \right)$ and increases as $x$ grows larger, tending towards infinity.

Would you like more details on the derivative analysis or other aspects of the graph?

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Here are some related questions to explore:

1. What is the first derivative of $f(x) = 9x^2 \ln(2x)$, and what does it tell us about the function’s increasing or decreasing behavior?
2. How does the second derivative help in determining the concavity of $f(x)$?
3. How would a vertical shift of the function affect its domain and range?
4. What are the asymptotic behaviors of similar logarithmic functions?
5. How can the concept of limits be used to understand the end behavior of this function?

Tip: Always check the domain of each component in a composite function to determine the overall domain accurately.