Chapter 1.2, Problem 44E

(a)

To determine

To find: The power function and the end behavior model for $f(x)$

Answer to Problem 44E

  $y=-x^2$ is the end behavior model.

Explanation of Solution

Given information: The function is $f(x)=\frac{-x^4+2x^2+x-3}{x^2-4}$

Calculation:

The given function is:

  $f(x)=\frac{-x^4+2x^2+x-3}{x^2-4}$

The ratio of the leading/(highest degree) terms of the numerator and denominator is the end behavior model for a rational function.

In the mentioned issue,

(1) The leading term in the numerator is $-x^4$

(2) The leading term in the denominator is $x^2$

Therefore, the end behavior model is

  $\begin{array}{c}y=\frac{-x^4}{x^2}\\ =-x^2\end{array}$

Therefore, the end behavior model is $y=-x^2$

(b)

To determine

To find: Any horizontal asymptotes.

Answer to Problem 44E

No horizontal asymptote.

Explanation of Solution

Given information: The function is $f(x)=\frac{-x^4+2x^2+x-3}{x^2-4}$

Calculation:

The ratio of the leading/(highest degree) terms of the numerator and denominator is the end behavior model for a rational function.

In the mentioned issue,

(1) The leading term in the numerator is $-x^4$

(2) The leading term in the denominator is $x^2$

Therefore, the end behavior model is

  $\begin{array}{c}y=\frac{-x^4}{x^2}\\ =-x^2\end{array}$

There are no horizontal asymptotes since the end behavior model is not a constant function.

Therefore, there is no horizontal asymptote.