To determine the most appropriate viewing window for graphing the function [tex]\( f(x) = \frac{x^2 - 81}{x^2 - 11x + 18} \)[/tex] on your graphing calculator, we need to understand the critical aspects of the function including its asymptotes, holes, and the general behavior of the function.
1. Factorizing the Denominator:
The denominator of our function is [tex]\( x^2 - 11x + 18 \)[/tex]. To find the zeros of the denominator, we can factorize as follows:
[tex]\[
x^2 - 11x + 18 = (x - 2)(x - 9)
\][/tex]
2. Holes of the Function:
Zeros of the denominator [tex]\( x = 2 \)[/tex] and [tex]\( x = 9 \)[/tex] would cause the function to be undefined at these points, leading to holes (or removable discontinuities) at these x-values.
3. Examining the Numerator:
The numerator is [tex]\( x^2 - 81 \)[/tex], which can be factorized as:
[tex]\[
x^2 - 81 = (x - 9)(x + 9)
\][/tex]
This means the numerator becomes zero at [tex]\( x = 9 \)[/tex] and [tex]\( x = -9 \)[/tex].
4. Simplification and Holes:
Since [tex]\( x = 9 \)[/tex] is a zero of both the numerator and denominator, this point is specifically a hole. The function simplifies to:
[tex]\[
f(x) = \frac{(x + 9)}{(x - 2)} \quad \text{for} \quad x \neq 9
\][/tex]
5. Asymptotes:
The vertical asymptote occurs where the function is undefined, excluding the holes. Therefore, [tex]\( x = 2 \)[/tex] is a vertical asymptote.
6. Behavior Analysis:
- As [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex]: The function approaches the broader behavior of [tex]\( \frac{x^2}{x^2} = 1 \)[/tex] indicating a horizontal asymptote at [tex]\( y = 1 \)[/tex].
Given this analysis, we need a viewing window that includes critical points like [tex]\( x = 2, 9 \)[/tex], and enough space to understand the behavior of the function around these points.
Considering the holes at [tex]\( x = 2 \)[/tex] and [tex]\( x = 9 \)[/tex], a good x-range should include these points and extend beyond to give a clear view of the behavior of the function.
The appropriate window is:
[tex]\[
X \min: -10, X \max: 10 \\
Y \min: -10, Y \max: 10
\][/tex]
This range provides a sufficiently broad view to include the key characteristics of the function, hence:
Answer: [tex]$X$[/tex] min: [tex]$-10$[/tex], [tex]$X$[/tex] max: 10 \\
[tex]$Y$[/tex] min: -10, [tex]$Y$[/tex] max: 10
