To find the image point of $(-4, -5)$ after the transformation $T_{4,2} \circ r_{y=x}$, we will follow these steps:

1. Apply the reflection $r_{y=x}$: The reflection across the line $y = x$ swaps the $x$ and $y$ coordinates of the point.

   $(-4, -5) \rightarrow (-5, -4)$

2. Apply the translation $T_{4,2}$: The translation $T_{4,2}$ adds 4 to the $x$ coordinate and 2 to the $y$ coordinate of the point.

   $(-5, -4) \rightarrow (-5 + 4, -4 + 2) = (-1, -2)$

Thus, the image point of $(-4, -5)$ after the transformation $T_{4,2} \circ r_{y=x}$ is $\boxed{(-1, -2)}$.

Would you like to go through any additional details or have more questions about this? Here are 5 related questions to consider:

1. What is the result if the reflection is across the $y$-axis instead?
2. How does the order of transformations affect the final result?
3. What is the geometric interpretation of $T_{4,2} \circ r_{y=x}$?
4. What happens if the point $(-4, -5)$ is first translated and then reflected?
5. How would the result change if the translation vector was $(4, -2)$ instead?

Tip: Always perform transformations in the correct order, as it can significantly change the outcome.