The given function is $f(x) = \ln(x^2) + 4x$, and the task is to determine if the inverse function $g(y)$ can be represented analytically.

Step-by-step reasoning:

1. Find the inverse function equation: To find $g(y)$, the function $f(x)$ must be solved for $x$ in terms of $y$: $y = \ln(x^2) + 4x$ Rearrange this equation to isolate $x$: $\ln(x^2) = y - 4x$ $x^2 = e^{y - 4x}$

2. Check if $x$ can be solved explicitly: The equation $x^2 = e^{y - 4x}$ involves both $x^2$ and $x$ within an exponential and logarithmic relationship, making it non-trivial to isolate $x$. Additionally, this type of equation typically cannot be solved in terms of elementary functions (logarithms, exponentials, powers, etc.).

3. Conclusion: Since $x$ cannot be explicitly expressed as a function of $y$ using standard analytical methods, the inverse function $g(y)$ is not analytically representable.

Final answer: Nein (No).

Would you like a detailed derivation of why it can't be solved analytically? Here are five related questions to deepen your understanding:

1. What conditions are necessary for a function to have an inverse?
2. How can the existence of an inverse function be verified using the horizontal line test?
3. What are elementary functions, and why do some equations resist being solved analytically?
4. How does the Lambert W function help solve specific equations involving exponentials and logarithms?
5. Can numerical methods approximate the value of the inverse function $g(y)$?

Tip: When working with equations involving both logarithms and exponentials, check for special functions like Lambert W or numerical solutions if an analytical one is not feasible.