To determine the domain of the function

$f(x) = 9x^2 \ln(2x)$

we need to analyze any restrictions on $x$ that would make $f(x)$ undefined.

Step 1: Identify Restrictions on $x$

1. Square Term, $9x^2$: There are no restrictions from $9x^2$, as this term is defined for all real $x$.

2. Logarithmic Term, $\ln(2x)$: The natural logarithm function $\ln(2x)$ is only defined when its argument $2x$ is positive: $2x > 0$ Solving this inequality: $x > 0$

Conclusion

The domain of $f(x) = 9x^2 \ln(2x)$ is all values of $x$ such that $x > 0$.

Domain in Interval Notation

The domain of $f(x)$ is: $(0, \infty)$

Answer

The correct answer is:

• $(0, \infty)$