The problem provides the equations of two functions:

1. $f(x) = -\frac{1}{4}x^3 + x + 5$
2. $g(x) = \frac{1}{2}x - 9$

The goal is to find where these two functions intersect, which occurs when $f(x) = g(x)$. To solve this, we need to equate the two functions and solve for $x$:

$-\frac{1}{4}x^3 + x + 5 = \frac{1}{2}x - 9$

Now, I will solve this equation for $x$.The equation $f(x) = g(x)$ has the following solutions:

1. $x = 4$ (real solution)
2. $x = -2 \pm 3.16i$ (complex solutions)

Since only real values of $x$ are relevant for the intersection point in the graph, the solution is $x = 4$.

So, $f(x) = g(x)$ when $x = 4$.

Would you like further details on this solution, or do you have any questions?

Here are 5 related questions to consider:

1. How do we graph cubic and linear functions to find intersections visually?
2. Can we verify the solution by substituting $x = 4$ into both functions?
3. What happens at complex solutions in terms of graph interpretation?
4. What is the behavior of cubic functions at their turning points?
5. How does the slope of a linear function affect where it intersects a cubic curve?

Tip: When solving cubic equations, always look for real solutions if you are working with graph intersections, as complex numbers don't represent points on a standard real-number graph.