Exercise 29 Page 760

We want to find the vertical and horizontal asymptotes in the graph of the function.

Let's start by factoring the denominator.

y = \frac{3 x ^{2} + 1}{x ^{2} + 2 x - 3}

Write as a difference

y = \frac{3 x ^{2} + 1}{x ^{2} + 3 x - x - 3}

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Factor out

Factor out $x$

y = \frac{3 x ^{2} + 1}{x ( x + 3 ) - x - 3}

Factor out $- 1$

y = \frac{3 x ^{2} + 1}{x ( x + 3 ) - 1 ( x + 3 )}

Factor out $(x + 3)$

y = \frac{3 x ^{2} + 1}{( x + 3 ) ( x - 1 )}

Recall that division by zero is not defined. Therefore, the rational function is undefined where

x + 3 = 0

and

x - 1 = 0 .

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