Precalculus Examples

$f (x) = \frac{9 x^{3} + 3 x^{2} - 2 x}{9 x^{2} + 3 x - 2}$

Step 1

Find where the expression $\frac{9 x^{3} + 3 x^{2} - 2 x}{9 x^{2} + 3 x - 2}$ is undefined.

$x = - \frac{2}{3}, x = \frac{1}{3}$

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 $m$ .

$n = 3$

$m = 2$

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 $x$ out of $9 x^{3} + 3 x^{2} - 2 x$ .

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Step 6.1.1.1.1

Factor $x$ out of $9 x^{3}$ .

$\frac{x (9 x^{2}) + 3 x^{2} - 2 x}{9 x^{2} + 3 x - 2}$

Step 6.1.1.1.2

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

$\frac{x (9 x^{2}) + x (3 x) - 2 x}{9 x^{2} + 3 x - 2}$

Step 6.1.1.1.3

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

$\frac{x (9 x^{2}) + x (3 x) + x \cdot - 2}{9 x^{2} + 3 x - 2}$

Step 6.1.1.1.4

Factor $x$ out of $x (9 x^{2}) + x (3 x)$ .

$\frac{x (9 x^{2} + 3 x) + x \cdot - 2}{9 x^{2} + 3 x - 2}$

Step 6.1.1.1.5

Factor $x$ out of $x (9 x^{2} + 3 x) + x \cdot - 2$ .

$\frac{x (9 x^{2} + 3 x - 2)}{9 x^{2} + 3 x - 2}$

Step 6.1.1.2

Factor by grouping.

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Step 6.1.1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot - 2 = - 18$ and whose sum is $b = 3$ .

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Step 6.1.1.2.1.1

Factor $3$ out of $3 x$ .

$\frac{x (9 x^{2} + 3 (x) - 2)}{9 x^{2} + 3 x - 2}$

Step 6.1.1.2.1.2

Rewrite $3$ as $- 3$ plus $6$

$\frac{x (9 x^{2} + (- 3 + 6) x - 2)}{9 x^{2} + 3 x - 2}$

Step 6.1.1.2.1.3

Apply the distributive property.

$\frac{x (9 x^{2} - 3 x + 6 x - 2)}{9 x^{2} + 3 x - 2}$

Step 6.1.1.2.2

Factor out the greatest common factor from each group.

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Step 6.1.1.2.2.1

Group the first two terms and the last two terms.

$\frac{x ((9 x^{2} - 3 x) + 6 x - 2)}{9 x^{2} + 3 x - 2}$

Step 6.1.1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (3 x (3 x - 1) + 2 (3 x - 1))}{9 x^{2} + 3 x - 2}$

Step 6.1.1.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$\frac{x ((3 x - 1) (3 x + 2))}{9 x^{2} + 3 x - 2}$

$\frac{x (3 x - 1) (3 x + 2)}{9 x^{2} + 3 x - 2}$

Step 6.1.2

Factor by grouping.

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Step 6.1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot - 2 = - 18$ and whose sum is $b = 3$ .

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Step 6.1.2.1.1

Factor $3$ out of $3 x$ .

$\frac{x (3 x - 1) (3 x + 2)}{9 x^{2} + 3 (x) - 2}$

Step 6.1.2.1.2

Rewrite $3$ as $- 3$ plus $6$

$\frac{x (3 x - 1) (3 x + 2)}{9 x^{2} + (- 3 + 6) x - 2}$

Step 6.1.2.1.3

Apply the distributive property.

$\frac{x (3 x - 1) (3 x + 2)}{9 x^{2} - 3 x + 6 x - 2}$

Step 6.1.2.2

Factor out the greatest common factor from each group.

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Step 6.1.2.2.1

Group the first two terms and the last two terms.

$\frac{x (3 x - 1) (3 x + 2)}{(9 x^{2} - 3 x) + 6 x - 2}$

Step 6.1.2.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 6.1.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

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

Step 6.1.3

Cancel the common factor of $3 x - 1$ .

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Step 6.1.3.1

Cancel the common factor.

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

Step 6.1.3.2

Rewrite the expression.

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

Step 6.1.4

Cancel the common factor of $3 x + 2$ .

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Step 6.1.4.1

Cancel the common factor.

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

Step 6.1.4.2

Divide $x$ by $1$ .

$x$

Step 6.2

Since there is no polynomial portion from the polynomial division, there are no oblique asymptotes.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

No Oblique Asymptotes

Step 8