The problem asks for the graph of the inverse of the function $f(x) = x^3 + 5x^2 + 1$, where $x \geq 0$. Let’s solve it step by step:

Key Steps:

1. Understand the function:

   • $f(x) = x^3 + 5x^2 + 1$ is a strictly increasing function for $x \geq 0$ (since its derivative $f'(x) = 3x^2 + 10x > 0$ for $x \geq 0$).
   • This means $f(x)$ is one-to-one and has an inverse on this domain.

2. Graphing the inverse:

   • The graph of an inverse function, $f^{-1}(x)$, is the reflection of the original function $f(x)$ across the line $y = x$.
   • We will look for a graph that satisfies this reflection symmetry for $x \geq 0$.

3. Analyze the given options:

   • The correct graph should:
     • Start near $(1, 0)$ because $f(0) = 1$, so the inverse $f^{-1}(1) = 0$.
     • Increase steadily and flatten out as $x$ grows, since $f(x)$ steepens in this range.

Observing the Options:

After comparing each graph against these criteria:

• The correct graph is the fourth option, which shows a reflection of $f(x)$ across $y = x$, starting near $(1, 0)$ and increasing.

Would you like further clarification or additional steps?

Follow-up Questions:

1. Why is $f(x) = x^3 + 5x^2 + 1$ one-to-one on $x \geq 0$?
2. How can you algebraically verify if a graph reflects across $y = x$?
3. What does the derivative $f'(x) = 3x^2 + 10x$ tell us about $f(x)$'s behavior?
4. How does restricting $x \geq 0$ affect the graph of the inverse?
5. Can you find the exact formula for $f^{-1}(x)$ algebraically?

Tip:

To visualize inverse functions, always check the symmetry of a function across $y = x$.