1. **Problem statement:** Find a rational function $f$ defined on its maximum domain that has a vertical asymptote at $x=7$. 2. **Recall:** A vertical asymptote occurs where the denominator of a rational function is zero but the numerator is not zero at that point. 3. **General form:** A rational function can be written as $$f(x) = \frac{P(x)}{Q(x)}$$ where $P(x)$ and $Q(x)$ are polynomials. 4. **Condition for vertical asymptote at $x=7$:** The denominator $Q(x)$ must be zero at $x=7$, so $Q(7) = 0$, and the numerator $P(7) \neq 0$. 5. **Example:** Choose $Q(x) = x - 7$ which is zero at $x=7$. 6. Choose $P(x) = 1$ which is never zero. 7. Thus, $$f(x) = \frac{1}{x - 7}$$ 8. **Domain:** All real numbers except $x=7$ where the function is undefined. 9. **Vertical asymptote:** At $x=7$ because the denominator is zero and numerator is nonzero. **Final answer:** $$f(x) = \frac{1}{x - 7}$$