QuestionAnalyze the function $R(x)=\frac{x^{4}+x^{2}+5}{x^{2}-9}$ .
A. Domain options:
1. $\{x \mid x \neq 3$ and $x \neq-3\}$
2. $\{x \mid x \neq 0\}$
3. $\{x \mid x \neq 0$ and $x \neq 3$ and $x \neq-3\}$
4. All real numbers
B. Vertical asymptote:
1. $x=3,-3$
2. No vertical asymptote.
C. Horizontal/oblique asymptote:
1. $y=$
2. No horizontal or oblique asymptote.

Studdy Solution

STEP 1

Assumptions1. The function is given by $R(x)=\frac{x^{4}+x^{}+5}{x^{}-9}$ . We are asked to find the domain of the function, the vertical asymptotes, and the horizontal or oblique asymptotes.

STEP 2

The domain of a function is the set of all possible input values (x-values) which will produce a valid output. For this rational function, the domain is all real numbers except for the values that make the denominator zero, since division by zero is undefined.

STEP 3

We find the values that make the denominator zero by setting the denominator equal to zero and solving for x.
$x^{2}-9=0$

STEP 4

This is a simple quadratic equation, which can be factored into $(x-3)(x+3)=0$ .

STEP 5

Setting each factor equal to zero gives the solutions $x=3$ and $x=-3$ .

STEP 6

Therefore, the domain of the function is all real numbers except $x=3$ and $x=-3$ . This corresponds to choice A $\{x \mid x \neq3$ and $x \neq-3\}$ .

STEP 7

Vertical asymptotes occur at the values of x that are not in the domain of the function. Therefore, the vertical asymptotes of the function are $x=3$ and $x=-3$ . This corresponds to choice A for the vertical asymptotes.

STEP 8

To find the horizontal or oblique asymptote of a rational function, we compare the degrees of the numerator and the denominator.

STEP 9

The degree of the numerator is4 (from the term $x^{4}$ ) and the degree of the denominator is2 (from the term $x^{2}$ ).

STEP 10

Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, there is an oblique asymptote.

STEP 11

To find the equation of the oblique asymptote, we perform long division of the numerator by the denominator.

STEP 12

Dividing $x^{4}$ by $x^{2}$ gives $x^{2}$ , and dividing $x^{2}$ by $x^{2}$ gives. The constant term5 in the numerator does not affect the oblique asymptote, so we ignore it.

STEP 13

Therefore, the equation of the oblique asymptote is $y=x^{2}+$ .

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Studdy Solution

STEP 1

Assumptions1. The function is given by $R(x)=\frac{x^{4}+x^{}+5}{x^{}-9}$ . We are asked to find the domain of the function, the vertical asymptotes, and the horizontal or oblique asymptotes.

STEP 2

The domain of a function is the set of all possible input values (x-values) which will produce a valid output. For this rational function, the domain is all real numbers except for the values that make the denominator zero, since division by zero is undefined.

STEP 3

We find the values that make the denominator zero by setting the denominator equal to zero and solving for x.
$x^{2}-9=0$

STEP 4

This is a simple quadratic equation, which can be factored into $(x-3)(x+3)=0$ .

STEP 5

Setting each factor equal to zero gives the solutions $x=3$ and $x=-3$ .

STEP 6

Therefore, the domain of the function is all real numbers except $x=3$ and $x=-3$ . This corresponds to choice A $\{x \mid x \neq3$ and $x \neq-3\}$ .

STEP 7

Vertical asymptotes occur at the values of x that are not in the domain of the function. Therefore, the vertical asymptotes of the function are $x=3$ and $x=-3$ . This corresponds to choice A for the vertical asymptotes.

STEP 8

To find the horizontal or oblique asymptote of a rational function, we compare the degrees of the numerator and the denominator.

STEP 9

The degree of the numerator is4 (from the term $x^{4}$ ) and the degree of the denominator is2 (from the term $x^{2}$ ).

STEP 10

Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, there is an oblique asymptote.

STEP 11

To find the equation of the oblique asymptote, we perform long division of the numerator by the denominator.

STEP 12

Dividing $x^{4}$ by $x^{2}$ gives $x^{2}$ , and dividing $x^{2}$ by $x^{2}$ gives. The constant term5 in the numerator does not affect the oblique asymptote, so we ignore it.

STEP 13

Therefore, the equation of the oblique asymptote is $y=x^{2}+$ .

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Studdy Solution

STEP 1

Assumptions1. The function is given by R(x)=x4+x+5x−9R(x)=\frac{x^{4}+x^{}+5}{x^{}-9}R(x)=x−9x4+x+5​ . We are asked to find the domain of the function, the vertical asymptotes, and the horizontal or oblique asymptotes.

STEP 2

The domain of a function is the set of all possible input values (x-values) which will produce a valid output. For this rational function, the domain is all real numbers except for the values that make the denominator zero, since division by zero is undefined.

STEP 3

We find the values that make the denominator zero by setting the denominator equal to zero and solving for x.x2−9=0x^{2}-9=0x2−9=0

STEP 4

This is a simple quadratic equation, which can be factored into (x−3)(x+3)=0(x-3)(x+3)=0(x−3)(x+3)=0.

STEP 5

Setting each factor equal to zero gives the solutions x=3x=3x=3 and x=−3x=-3x=−3.

STEP 6

Therefore, the domain of the function is all real numbers except x=3x=3x=3 and x=−3x=-3x=−3. This corresponds to choice A {x∣x≠3\{x \mid x \neq3{x∣x=3 and x≠−3}x \neq-3\}x=−3}.

STEP 7

Vertical asymptotes occur at the values of x that are not in the domain of the function. Therefore, the vertical asymptotes of the function are x=3x=3x=3 and x=−3x=-3x=−3. This corresponds to choice A for the vertical asymptotes.

STEP 8

To find the horizontal or oblique asymptote of a rational function, we compare the degrees of the numerator and the denominator.

STEP 9

The degree of the numerator is4 (from the term x4x^{4}x4) and the degree of the denominator is2 (from the term x2x^{2}x2).

STEP 10

Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, there is an oblique asymptote.

STEP 11

To find the equation of the oblique asymptote, we perform long division of the numerator by the denominator.

STEP 12

Dividing x4x^{4}x4 by x2x^{2}x2 gives x2x^{2}x2, and dividing x2x^{2}x2 by x2x^{2}x2 gives. The constant term5 in the numerator does not affect the oblique asymptote, so we ignore it.

STEP 13

Therefore, the equation of the oblique asymptote is y=x2+y=x^{2}+y=x2+.

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Assumptions1. The function is given by $R(x)=\frac{x^{4}+x^{}+5}{x^{}-9}$ . We are asked to find the domain of the function, the vertical asymptotes, and the horizontal or oblique asymptotes.

We find the values that make the denominator zero by setting the denominator equal to zero and solving for x.
$x^{2}-9=0$

This is a simple quadratic equation, which can be factored into $(x-3)(x+3)=0$ .

Setting each factor equal to zero gives the solutions $x=3$ and $x=-3$ .

Therefore, the domain of the function is all real numbers except $x=3$ and $x=-3$ . This corresponds to choice A $\{x \mid x \neq3$ and $x \neq-3\}$ .

Vertical asymptotes occur at the values of x that are not in the domain of the function. Therefore, the vertical asymptotes of the function are $x=3$ and $x=-3$ . This corresponds to choice A for the vertical asymptotes.

The degree of the numerator is4 (from the term $x^{4}$ ) and the degree of the denominator is2 (from the term $x^{2}$ ).

Dividing $x^{4}$ by $x^{2}$ gives $x^{2}$ , and dividing $x^{2}$ by $x^{2}$ gives. The constant term5 in the numerator does not affect the oblique asymptote, so we ignore it.

Therefore, the equation of the oblique asymptote is $y=x^{2}+$ .