How do you sketch the curve for $y = \frac{x^{2} + 1}{x^{2} - 4}$ $y= (x^2+1)/(x^2-4)$ ?

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How do you sketch the curve for $y = \frac{x^{2} + 1}{x^{2} - 4}$ $y= (x^2+1)/(x^2-4)$ ?

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Answer 1 · 2015-08-02T22:08:05

Find the critical numbers, local maximum and determine the behaviors as $\pm \infty$ $+-oo$ .

Explanation:

Step 1. Determine the domain and range of $f (x) = \frac{x^{2} + 1}{x^{2} - 4}$ $f(x)=(x^2+1)/(x^2-4)$ .

Domain is all values of $x$ $x$ except where
$x^{2} - 4 = 0$ $x^2-4=0$
$x^{2} = 4$ $x^2=4$
$x = \pm 2$ $x=+-2$

Range is $(- \infty, \infty)$ $(-oo,oo)$

Step 2. Find the critical numbers of $f (x)$ $f(x)$ in $(- 2, 2)$ $(-2,2)$ .
Apply the Quotient Rule to differentiate the function.
$f'(x)=((x^2-4)(2x)-(x^2+1)(2x))/(x^2-4)^2$
$f'(x)=(2x)/(x^2-4)-(2x(x^2+1))/(x^2-4)^2$

Critical numbers:
$x=+-2$ , where $f'(x)$ does not exist, and $x=0$ , where $f'(x)=0$ .

Step 3. Calculate $f(x)$ with critical numbers.
$f(-2)$ does not exist. The limit from the left is $+oo$ and the limit from the right is $-oo$ .
$f(2)$ does not exist. The limit from the left is $-oo$ and the limit from the right is $+oo$ .
$f(0) = -1/4$ is a local maximum.
For $x<-2$ , $f(x)>0$ . For $x>2$ , $f(x)>0$ .

Step 4. Sketch the graph with these attributes.

graph{(x^2+1)/(x^2-4) [-20, 20, -10, 10]}

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How do you sketch the curve for $y = \frac{x^{2} + 1}{x^{2} - 4}$ $y= (x^2+1)/(x^2-4)$ ?

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Explanation:

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How do you sketch the curve for $y = \frac{x^{2} + 1}{x^{2} - 4}$ $y= (x^2+1)/(x^2-4)$ ?