Question #41554

This question tests the student's ability to handle the intersection of a function and its inverse on a restricted domain, combined with inverse function integration.

While students correctly identify that $f(x) = f^{-1}(x)$ simplifies to $f(x) = x$ for strictly increasing functions, they frequently fail to cross-reference the resulting roots with the function's domain.

Here, solving $-x^2 + 6x - 4 = x$ yields $x=1$ and $x=4$.

The primary trap is the extraneous root $x=4$, which lies outside the domain $(-\infty, 3]$.

Using $x=4$ as the upper limit leads directly to the incorrect distractor option 20/3.

Furthermore, the problem rigorously tests the application of the inverse integration formula $\int f^{-1}(x)dx = xf^{-1}(x) - \int f(y)dy$, requiring precise mapping of the boundary conditions to avoid sign errors.