Explore IDNLearn.com to discover insightful answers from experts and enthusiasts alike. Join our Q&A platform to access reliable and detailed answers from experts in various fields.
Sagot :
To determine the end behavior of the function [tex]\( f(x) = \frac{x^2 - 4}{x^2 - 9} \)[/tex], we need to consider the function as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\(\infty\)[/tex]. The end behavior of a rational function is typically analyzed by examining the degrees of the polynomial in the numerator and the polynomial in the denominator.
Here, both the numerator ([tex]\(x^2 - 4\)[/tex]) and the denominator ([tex]\(x^2 - 9\)[/tex]) are quadratic polynomials, meaning both have the highest degree term of [tex]\(x^2\)[/tex].
For very large positive or negative values of [tex]\(x\)[/tex], the lower-degree terms (constants in this case) become negligible in comparison to the higher-degree terms. We can thus simplify the analysis by focusing on the highest degree terms from both the numerator and the denominator.
The function can be approximated by:
[tex]\[ f(x) \approx \frac{x^2}{x^2}. \][/tex]
Simplifying this, we get:
[tex]\[ f(x) \approx 1 \][/tex]
Therefore, as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\(\infty\)[/tex], the function [tex]\( f(x) \)[/tex] approaches 1.
So, the correct answer is:
D. The function approaches 1 as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\(\infty\)[/tex].
Here, both the numerator ([tex]\(x^2 - 4\)[/tex]) and the denominator ([tex]\(x^2 - 9\)[/tex]) are quadratic polynomials, meaning both have the highest degree term of [tex]\(x^2\)[/tex].
For very large positive or negative values of [tex]\(x\)[/tex], the lower-degree terms (constants in this case) become negligible in comparison to the higher-degree terms. We can thus simplify the analysis by focusing on the highest degree terms from both the numerator and the denominator.
The function can be approximated by:
[tex]\[ f(x) \approx \frac{x^2}{x^2}. \][/tex]
Simplifying this, we get:
[tex]\[ f(x) \approx 1 \][/tex]
Therefore, as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\(\infty\)[/tex], the function [tex]\( f(x) \)[/tex] approaches 1.
So, the correct answer is:
D. The function approaches 1 as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\(\infty\)[/tex].
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Thank you for choosing IDNLearn.com. We’re committed to providing accurate answers, so visit us again soon.