Question: The graph of the reciprocal parent function, $f(x) = \frac{1}{x}$, is shifted 8 units down and 6 units to the left to create the graph of $g(x)$. What function is $g(x)$?
1. **State the problem:** We start with the reciprocal parent function $$f(x) = \frac{1}{x}$$ and apply two transformations: shifting 8 units down and 6 units to the left.
2. **Recall the transformation rules:**
- Shifting a graph $h$ units to the left means replacing $x$ with $x + h$ in the function.
- Shifting a graph $k$ units down means subtracting $k$ from the entire function.
3. **Apply the horizontal shift:**
Since the graph is shifted 6 units to the left, replace $x$ with $x + 6$:
$$f(x) = \frac{1}{x} \to \frac{1}{x + 6}$$
4. **Apply the vertical shift:**
Since the graph is shifted 8 units down, subtract 8 from the function:
$$g(x) = \frac{1}{x + 6} - 8$$
5. **Final function:**
$$g(x) = \frac{1}{x + 6} - 8$$
6. **Match with the options:**
Option C is $$g(x) = \frac{1}{x + 6} - 8$$ which matches our result.
**Answer:** Option C