Answered: For the given rational function…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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For the given rational function

f(x)=6−2x^2/x^2−9,

(A) Find any intercepts for the graph.

(B) Find any vertical and horizontal asymptotes for the graph.

Expert Solution

Step 1

Consider the given function $f (x) = \frac{6 - 2 x^{2}}{x^{2} - 9}$ .

Let $y = \frac{6 - 2 x^{2}}{x^{2} - 9}$ .

(A) Intercepts:

Obtain the y-intercept of the given equation by substituting x = 0 in $y = \frac{6 - 2 x^{2}}{x^{2} - 9}$ .

$\begin{array}{rcl} y & = & \frac{6 - 2 {(0)}^{2}}{{(0)}^{2} - 9} \\ = & - \frac{6}{9} \\ = & - \frac{2}{3} \end{array}$

That is, the y-intercept is $\begin{array}{rcl} (0, - \frac{2}{3}) \end{array}$ .

Obtain the x-intercept of the given equation by substituting y = 0 in $y = \frac{6 - 2 x^{2}}{x^{2} - 9}$ .

$0 = \frac{6 - 2 x^{2}}{x^{2} - 9} 6 - 2 x^{2} = 0 - 2 x^{2} = - 6 x^{2} = 3 x = \pm \sqrt{3}$

That is, the x-intercepts are $(\sqrt{3}, 0) and (- \sqrt{3}, 0)$ .

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For the given rational function f(x)=6−2x^2/x^2−9​, ​(A) Find any intercepts for the graph. ​(B) Find any vertical and horizontal asymptotes for the graph. ​(C) Graph the function in a standard viewing window using a graphing calculator.

For the given rational function f(x)=6−2x^2/x^2−9, (A) Find any intercepts for the graph. (B) Find any vertical and horizontal asymptotes for the graph. (C) Graph the function in a standard viewing window using a graphing calculator.