IDNLearn.com connects you with a community of experts ready to answer your questions. Ask your questions and receive reliable and comprehensive answers from our dedicated community of professionals.

Select the correct answer.

Which statement describes the end behavior of the function?

[tex]
f(x) = \frac{x^2 - 4}{x^2 - 9}
[/tex]

A. The function approaches 0 as [tex]x[/tex] approaches [tex]-\infty[/tex] and [tex]\infty[/tex].

B. The function approaches [tex]\frac{4}{9}[/tex] as [tex]x[/tex] approaches [tex]-\infty[/tex] and [tex]\infty[/tex].

C. The function approaches [tex]\frac{2}{3}[/tex] as [tex]x[/tex] approaches [tex]-\infty[/tex] and [tex]\infty[/tex].

D. The function approaches 1 as [tex]x[/tex] approaches [tex]-\infty[/tex] and [tex]\infty[/tex].


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].