Trigonometry Examples

$f (x) = 0.8 x + 2.9 f^{- 1}$

Step 1

Find where the expression $0.8 x + \frac{2.9}{f}$ is undefined.

$x = 0$

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 $m$ .

$n = 1$

$m = 0$

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

Combine.

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Step 6.1.1

To write $0.8 x$ as a fraction with a common denominator, multiply by $\frac{f}{f}$ .

$\frac{0.8 x f}{f} + \frac{2.9}{f}$

Step 6.1.2

Combine the numerators over the common denominator.

$\frac{0.8 x f + 2.9}{f}$

Step 6.1.3

Factor $0.1$ out of $0.8 x f + 2.9$ .

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Step 6.1.3.1

Move $x$ .

$\frac{0.8 f x + 2.9}{f}$

Step 6.1.3.2

Factor $0.1$ out of $0.8 f x$ .

$\frac{0.1 (8 f x) + 2.9}{f}$

Step 6.1.3.3

Factor $0.1$ out of $2.9$ .

$\frac{0.1 (8 f x) + 0.1 (29)}{f}$

Step 6.1.3.4

Factor $0.1$ out of $0.1 (8 f x) + 0.1 (29)$ .

$\frac{0.1 (8 f x + 29)}{f}$

Step 6.1.4

Simplify.

$\frac{0.8 f x + 2.9}{f}$

Step 6.2

Factor $0.1$ out of $0.8 f x + 2.9$ .

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Step 6.2.1

Factor $0.1$ out of $0.8 f x$ .

$\frac{0.1 (8 f x) + 2.9}{f}$

Step 6.2.2

Factor $0.1$ out of $2.9$ .

$\frac{0.1 (8 f x) + 0.1 (29)}{f}$

Step 6.2.3

Factor $0.1$ out of $0.1 (8 f x) + 0.1 (29)$ .

$\frac{0.1 (8 f x + 29)}{f}$

Step 6.3

Expand $0.1 (8 f x + 29)$ .

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Step 6.3.1

Apply the distributive property.

$\frac{0.1 (8 f x) + 0.1 \cdot 29}{f}$

Step 6.3.2

Move parentheses.

$\frac{0.1 (8 f) x + 0.1 \cdot 29}{f}$

Step 6.3.3

Remove parentheses.

$\frac{0.1 \cdot (8 f x) + 0.1 \cdot 29}{f}$

Step 6.3.4

Multiply $0.1$ by $8$ .

$\frac{0.8 f x + 0.1 \cdot 29}{f}$

Step 6.3.5

Multiply $0.1$ by $29$ .

$\frac{0.8 f x + 2.9}{f}$

Step 6.4

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

$f$

$0$

$0.8 f x$

$2.9$

Step 6.5

Divide the highest order term in the dividend $0.8 f x$ by the highest order term in divisor $f$ .

					$0.8 x$
	$f$	+	$0$		$0.8 f x$	+	$2.9$

Step 6.6

Multiply the new quotient term by the divisor.

				$0.8 x$
$f$	+	$0$		$0.8 f x$	+	$2.9$
			+	$0.8 x f$	+	$0$

Step 6.7

The expression needs to be subtracted from the dividend, so change all the signs in $0.8 x f + 0$

				$0.8 x$
$f$	+	$0$		$0.8 f x$	+	$2.9$
			-	$0.8 x f$	-	$0$

Step 6.8

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

				$0.8 x$
$f$	+	$0$		$0.8 f x$	+	$2.9$
			-	$0.8 x f$	-	$0$
					+	$2.9$

Step 6.9

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

$0.8 x + \frac{2.9}{f}$

Step 6.10

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

$y = 0.8 x$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = 0.8 x$

Step 8