Rational Functions Math Example 3

Follow the full solution, then compare it with the other examples linked below.

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Find the

x

-intercepts and vertical asymptotes of

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

1
Factor: $g(x) = \frac{(x-1)(x+1)}{(x+1)(x+2)}$ . Cancel $(x+1)$ : $g(x) = \frac{x-1}{x+2}$ for $x \neq -1$ .
2
$x$ -intercept: set numerator $= 0$ : $x - 1 = 0$ , so $x = 1$ . Vertical asymptote: $x + 2 = 0$ , so $x = -2$ . Hole at $x = -1$ .

Answer

x\text{-intercept: } (1, 0), \quad \text{VA: } x = -2, \quad \text{Hole at } x = -1

Always factor and simplify first to distinguish holes from vertical asymptotes. The

x

-intercepts come from the simplified numerator.

About Rational Functions

A rational function is a ratio of two polynomials: $f(x) = P(x)/Q(x)$ where $P$ and $Q$ are polynomials and $Q(x) \neq 0$ .

Find the vertical and horizontal asymptotes of [formula].

Find all asymptotes and holes of [formula].

Find all asymptotes (vertical, horizontal, and oblique) of [formula]. Does the function have a hole