Algebra Examples

$y = x - \frac{1}{x}$

Step 1

Find where the expression $x - \frac{1}{x}$ is undefined.

$x = 0$

Step 2

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 3

Find $n$ and $m$ .

$n = 2$

$m = 1$

Step 4

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

No Horizontal Asymptotes

Step 5

Find the oblique asymptote using polynomial division.

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Step 5.1

Combine.

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Step 5.1.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x \cdot x}{x} - \frac{1}{x}$

Step 5.1.2

Combine the numerators over the common denominator.

$\frac{x \cdot x - 1}{x}$

Step 5.1.3

Simplify the numerator.

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Step 5.1.3.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} x - 1}{x}$

Step 5.1.3.2

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{1} - 1}{x}$

Step 5.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 1} - 1}{x}$

Step 5.1.3.4

Add $1$ and $1$ .

$\frac{x^{2} - 1}{x}$

Step 5.1.3.5

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

$\frac{x^{2} - 1^{2}}{x}$

Step 5.1.3.6

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 $b = 1$ .

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

Step 5.1.4

Simplify.

$\frac{x^{2} - 1}{x}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

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

$\frac{x^{2} - 1^{2}}{x}$

Step 5.2.2

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 $b = 1$ .

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

Step 5.3

Expand $(x + 1) (x - 1)$ .

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Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 1 + 1 (x - 1)}{x}$

Step 5.3.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1}{x}$

Step 5.3.4

Reorder $x$ and $- 1$ .

$\frac{x \cdot x - 1 x + 1 x + 1 \cdot - 1}{x}$

Step 5.3.5

Raise $x$ to the power of $1$ .

$\frac{x \cdot x - 1 x + 1 x + 1 \cdot - 1}{x}$

Step 5.3.6

Raise $x$ to the power of $1$ .

$\frac{x \cdot x - 1 x + 1 x + 1 \cdot - 1}{x}$

Step 5.3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 1} - 1 x + 1 x + 1 \cdot - 1}{x}$

Step 5.3.8

Add $1$ and $1$ .

$\frac{x^{2} - 1 x + 1 x + 1 \cdot - 1}{x}$

Step 5.3.9

Multiply $x$ by $1$ .

$\frac{x^{2} - 1 x + x + 1 \cdot - 1}{x}$

Step 5.3.10

Multiply $- 1$ by $1$ .

$\frac{x^{2} - 1 x + x - 1}{x}$

Step 5.3.11

Add $- 1 x$ and $x$ .

$\frac{x^{2} + 0 - 1}{x}$

Step 5.3.12

Subtract $1$ from $0$ .

$\frac{x^{2} - 1}{x}$

Step 5.4

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

$x$

$0$

$x^{2}$

$0 x$

$1$

Step 5.5

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x$ .

					$x$
	$x$	+	$0$		$x^{2}$	+	$0 x$	-	$1$

Step 5.6

Multiply the new quotient term by the divisor.

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	-	$1$
			+	$x^{2}$	+	$0$

Step 5.7

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 0$

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	-	$1$
			-	$x^{2}$	-	$0$

Step 5.8

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

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	-	$1$
			-	$x^{2}$	-	$0$
						$0$

Step 5.9

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

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	-	$1$
			-	$x^{2}$	-	$0$
						$0$	-	$1$

Step 5.10

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

$x - \frac{1}{x}$

Step 5.11

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

$y = x$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x$

Step 7