For the function Upper F left parenthesis x right parenthesis equals StartFraction 1 Over left parenthesis x minus 9 right parenthesis squared EndFraction
​, ​(a) graph the rational function using​ transformations, (b) use the final graph to find the domain and​ range, and​ (c) use the final graph to list any​ vertical, horizontal, or oblique asymptotes. What is the domain of the given​ function?

Let's analyze the function \( F(x) = \frac{1}{(x - 9)^2} \) step by step to answer the questions.

### (a) Graph the rational function using transformations
We start with the basic function \( f(x) = \frac{1}{x^2} \), which has the following characteristics:
- It's always positive for all \( x \neq 0 \), with a vertical asymptote at \( x = 0 \).
- As \( x \to 0 \), the function approaches \( +\infty \).
- As \( x \to \pm \infty \), the function approaches \( 0 \) (horizontal asymptote at \( y = 0 \)).

#### Transformations applied to \( f(x) = \frac{1}{x^2} \):
1. **Horizontal shift**: \( F(x) = \frac{1}{(x - 9)^2} \) shifts the graph of \( \frac{1}{x^2} \) **9 units to the right**. So, the vertical asymptote moves to \( x = 9 \).

This gives us the graph of \( F(x) \). It retains the same shape as \( f(x) = \frac{1}{x^2} \), but it is centered at \( x = 9 \).

### (b) Use the final graph to find the domain and range

#### Domain:
The domain of \( F(x) = \frac{1}{(x - 9)^2} \) is all real numbers except where the denominator is zero. The denominator is zero when \( x = 9 \), so the function is undefined there.
Thus, the **domain** is:
\[ \boxed{(-\infty, 9) \cup (9, +\infty)} \]

#### Range:
Since \( (x - 9)^2 \) is always positive for \( x \neq 9 \), and the function \( \frac{1}{(x - 9)^2} \) outputs positive values, the graph of \( F(x) \) stays above the x-axis, approaching zero but never reaching it.
Thus, the **range** is:
\[ \boxed{(0, +\infty)} \]

### (c) Asymptotes

#### Vertical Asymptote:
- There is a vertical asymptote at \( x = 9 \), as the function tends to infinity as \( x \to 9 \).

#### Horizontal Asymptote:
- As \( x \to \pm \infty \), \( F(x) = \frac{1}{(x - 9)^2} \to 0 \), so there is a **horizontal asymptote** at \( y = 0 \).

#### Oblique Asymptote:
- There are **no oblique asymptotes** because the degree of the numerator is less than the degree of the denominator.

### Summary:
1. **Domain**: \( (-\infty, 9) \cup (9, +\infty) \)
2. **Range**: \( (0, +\infty) \)
3. **Vertical Asymptote**: \( x = 9 \)
4. **Horizontal Asymptote**: \( y = 0 \)
5. **No oblique asymptote**

Would you like a detailed graph of the function?

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Here are 5 questions to explore:
1. What is the behavior of the function near \( x = 9 \)?
2. How does the transformation affect the graph's symmetry?
3. Can you determine the limits as \( x \to 9^- \) and \( x \to 9^+ \)?
4. What happens if we change the power in the denominator from 2 to 1?
5. How would a vertical shift affect the graph?

**Tip**: Asymptotes give critical insights into the long-term behavior of a function! Always check for them when dealing with rational functions.

For the function F(x) = 1 / (x - 9)^2, (a) graph the rational function using transformations, (b) use the final graph to find the domain and range, and (c) use the final graph to list any vertical, horizontal, or oblique asymptotes. What is the domain of the given function?

Learn how to graph the rational function F(x) = 1 / (x - 9)^2 using transformations. This guide explains the domain, range, and asymptotes, with detailed steps on graphing and identifying vertical and horizontal asymptotes. Ideal for Grades 10-12 math students.

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