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The problem mentions the function \( f(x) \) as the "rational parent function," which is typically interpreted as:

\[
f(x) = \frac{1}{x}
\]

It also provides another function \( g(x) = x - 7 \), and we are asked to analyze the horizontal asymptote of the composition \( g(f(x)) \).

### Step-by-step breakdown:

1. **Parent Rational Function**: \( f(x) = \frac{1}{x} \) is a rational function. The horizontal asymptote of this parent function is \( y = 0 \), because as \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to 0 \).

2. **Composition of Functions**: Now, we are looking at \( g(f(x)) \), which is:

   \[
   g(f(x)) = g\left( \frac{1}{x} \right) = \frac{1}{x} - 7
   \]

3. **Horizontal Asymptote of \( g(f(x)) \)**: To find the horizontal asymptote of this new function, we analyze its behavior as \( x \to \infty \) or \( x \to -\infty \).
   
   - As \( x \to \infty \), \( \frac{1}{x} \to 0 \). Thus, \( g(f(x)) \to 0 - 7 = -7 \).
   - As \( x \to -\infty \), the same thing happens: \( g(f(x)) \to -7 \).

   Hence, the horizontal asymptote of \( g(f(x)) \) is \( y = -7 \).

### Final Answer:
The horizontal asymptote of \( g(f(x)) \) is \( y = -7 \).

Would you like help graphing this function or explaining further details?

Here are 5 related questions to deepen your understanding:
1. What is a rational function, and how do we find its asymptotes?
2. How do we handle transformations like shifting or scaling in rational functions?
3. Can vertical asymptotes change when composing functions like in this problem?
4. What is the domain of \( g(f(x)) \) in this example?
5. How does the graph of \( f(x) = \frac{1}{x} \) differ from other rational functions?

**Tip**: When composing functions, pay close attention to how transformations (like translations) affect both vertical and horizontal asymptotes.

Let f(x) be the rational parent function and g(x) = x - 7. Use the grid to graph the horizontal asymptote of g(f(x)).

Explore how to find and graph the horizontal asymptote of the function g(f(x)) where f(x) = 1/x and g(x) = x - 7. This solution includes step-by-step analysis of function composition and asymptote behavior for high school level math.

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