Question 1189261
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If you want to use a graphing utility to draw the graph, as suggested by the other tutor, thereby learning absolutely nothing about rational functions and asymptotes, then that's one choice you have.<br>
On the other hand, if you want to learn something from this problem....<br>
{{{R(x)=(x^2-1)/(x^4-16)}}}<br>
Factor the numerator and denominator into linear factors as much as possible:<br>
{{{R(x)=((x+1)(x-1))/((x+2)(x-2)(x^2+4))}}}<br>
Linear factors in the numerator tell you where the zeros of the function are; linear factors in the denominator tell you where the asymptotes are.<br>
This function has zeros at x = -1 and x = 1; it has asymptotes as x = -2 and x=2.<br>
To analyze the function, imagine "walking" along the x-axis and observing when the function value is positive and when it is negative.  With the numerator and denominator in factored form, it is easy to see that, as you walk along the x-axis, the only places where the sign of the function can change are where the zeros and asymptotes are.<br>
When you pass a zero of the function, the value of the function changes sign and the graph is continuous.<br>
When you pass an x value where there is an asymptote, the sign of the function changes and the graph is not continuous.  When you are close to an asymptote, you are dividing by a number that is close to zero; that means the absolute value of the function is very large.  So when you pass an asymptote, the function value either changes from large negative to large positive, or from large positive to large negative.<br>
So with that background we can begin to analyze the function.<br>
One easy place to start is at x=0:<br>
{{{R(0)=-1/-16 = 1/16}}}<br>
So the function value is positive at x=0.<br>
Now "walk" to the right from 0.  At x=1 the function has a zero, so the sign of the function changes to negative; the graph is continuous and passes through the x-axis.<br>
Then at x=2 there is an asymptote, so the function changes sign and the graph is discontinuous.  Since the function value was negative before we got to x=2, the function value changes from large negative to large positive when we pass x=2.<br>
Then perform the similar analysis walking to the left starting at x=0.  You will see the function value changes from positive to negative and the graph is continuous at x=-1, and the function value changes from large negative to large positive at x=-2.<br>
NOW we should graph the function and verify that our analysis is correct:<br>
{{{graph(400,400,-3,3,-1,1,(x^2-1)/(x^4-16))}}}<br>