Precalculus Examples

$f (x) = \frac{3 (x - 1) (x + 1)}{2 (x - 1)}$

Step 1

Find where the expression $\frac{3 (x - 1) (x + 1)}{2 (x - 1)}$ is undefined.

$x = 1$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 4

Find $n$ and $m$ .

$n = 2$

$m = 1$

Step 5

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

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Step 6.1

Simplify the expression.

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Step 6.1.1

Simplify the numerator.

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Step 6.1.1.1

Factor $3$ out of $3 x^{2} - 3$ .

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Step 6.1.1.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2}) - 3}{2 x - 2}$

Step 6.1.1.1.2

Factor $3$ out of $- 3$ .

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

Step 6.1.1.1.3

Factor $3$ out of $3 (x^{2}) + 3 (- 1)$ .

$\frac{3 (x^{2} - 1)}{2 x - 2}$

Step 6.1.1.2

Rewrite $1$ as $1^{2}$ .

$\frac{3 (x^{2} - 1^{2})}{2 x - 2}$

Step 6.1.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{3 (x + 1) (x - 1)}{2 x - 2}$

Step 6.1.2

Simplify terms.

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Step 6.1.2.1

Factor $2$ out of $2 x - 2$ .

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Step 6.1.2.1.1

Factor $2$ out of $2 x$ .

$\frac{3 (x + 1) (x - 1)}{2 (x) - 2}$

Step 6.1.2.1.2

Factor $2$ out of $- 2$ .

$\frac{3 (x + 1) (x - 1)}{2 (x) + 2 (- 1)}$

Step 6.1.2.1.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$\frac{3 (x + 1) (x - 1)}{2 (x - 1)}$

Step 6.1.2.2

Cancel the common factor of $x - 1$ .

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Step 6.1.2.2.1

Cancel the common factor.

$\frac{3 (x + 1) (x - 1)}{2 (x - 1)}$

Step 6.1.2.2.2

Rewrite the expression.

$\frac{3 (x + 1)}{2}$

Step 6.2

Expand $3 (x + 1)$ .

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Step 6.2.1

Apply the distributive property.

$\frac{3 x + 3 \cdot 1}{2}$

Step 6.2.2

Multiply $3$ by $1$ .

$\frac{3 x + 3}{2}$

Step 6.3

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$2$

$3 x$

$3$

Step 6.4

Divide the highest order term in the dividend $3 x$ by the highest order term in divisor $2$ .

			$\frac{3 x}{2}$
	$2$		$3 x$	+	$3$

Step 6.5

Multiply the new quotient term by the divisor.

		$\frac{3 x}{2}$
$2$		$3 x$	+	$3$
	+	$3 x$

Step 6.6

The expression needs to be subtracted from the dividend, so change all the signs in $3 x$

		$\frac{3 x}{2}$
$2$		$3 x$	+	$3$
	-	$3 x$

Step 6.7

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

		$\frac{3 x}{2}$
$2$		$3 x$	+	$3$
	-	$3 x$
		$0$

Step 6.8

Pull the next terms from the original dividend down into the current dividend.

		$\frac{3 x}{2}$
$2$		$3 x$	+	$3$
	-	$3 x$
		$0$	+	$3$

Step 6.9

The final answer is the quotient plus the remainder over the divisor.

$\frac{3 x}{2} + \frac{3}{2}$

Step 6.10

The oblique asymptote is the polynomial portion of the long division result.

$y = \frac{3 x}{2}$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{3 x}{2}$

Step 8