To solve this problem, we need to determine how the given function \( f(x) = \frac{a-19}{x} + 5 \) transforms when the graph is translated 3 units down and 4 units to the right to produce the graph of \( y = g(x) \).
### Step-by-Step Solution:
1. Translation Basics:
- Translating a graph to the right by \( h \) units implies substituting \( x \) with \( (x - h) \) in the function.
- Translating a graph downward by \( k \) units means subtracting \( k \) from the entire function.
2. Translate 4 Units to the Right:
- To move the graph 4 units to the right, we substitute \( x \) with \((x - 4)\) in \( f(x) \):
[tex]\[
f(x - 4) = \frac{a-19}{x-4} + 5
\][/tex]
3. Translate 3 Units Down:
- To move the graph 3 units down, we subtract 3 from the result of the previous step:
[tex]\[
g(x) = \left(\frac{a-19}{x-4} + 5\right) - 3 = \frac{a-19}{x-4} + (5 - 3)
\][/tex]
4. Simplify the Translated Function:
- Simplify the expression within the parentheses:
[tex]\[
g(x) = \frac{a-19}{x-4} + 2
\][/tex]
This matches option (B):
[tex]\[
g(x) = \frac{a-19}{x-4} + 2
\][/tex]
5. Verify Against Other Options:
- Option (A): \( g(x) = \frac{a-19}{x+4} + 2 \)
- This is incorrect because the \( x \) in the denominator should be \( (x - 4) \), not \( (x + 4) \).
- Option (C): \( g(x) = \frac{a-22}{x+4} + 5 \)
- This is also incorrect because neither the numerator \((a-22)\) nor the \( x \) term in the denominator \((x+4)\) matches the required transformations.
Therefore, the correct equation defining the function \( g(x) \) is:
[tex]\[
\boxed{B}
\][/tex]
