Question 177·Hard·Nonlinear Functions

The function $f$ is defined by

In the $xy$-plane, the graph of $y = h(x)$ is obtained by performing the following transformations on the graph of $y = f(x)$, in the order given:

1. Reflect the graph across the $y$-axis.
2. Translate the resulting graph $3$ units to the right.
3. Translate the resulting graph $7$ units down.

What is the value of $h(2)$?