Chapter 1.2, Problem 41E

(a)

To determine

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

Answer to Problem 41E

  $y=1/2x$ is the end behavior model.

Explanation of Solution

Given information: The function is $f(x)=\frac{x-2}{2x^2+3x-5}$

Calculation:

The given function is:

  $f(x)=\frac{x-2}{2x^2+3x-5}$

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 $x$

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

Therefore, the end behavior model is

  $\begin{array}{c}y=\frac{x}{2x^2}\\ =\frac{1}{2x}\end{array}$

Therefore, the end behavior model is $y=1/2x$ .

(b)

To determine

To find: Any horizontal asymptotes.

Answer to Problem 41E

No horizontal asymptote.

Explanation of Solution

Given information: The function is $f(x)=\frac{x-2}{2x^2+3x-5}$

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 $x$

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

Therefore, the end behavior model is

  $\begin{array}{c}y=\frac{x}{2x^2}\\ =\frac{1}{2x}\end{array}$

Therefore, the end behavior model is $y=1/2x$ .

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

Therefore, there is no horizontal asymptote.