Questions: The rational function R(x)=(x^3-8)/x has a/an [Select ] " asymptote. Choose the best answer. [Select] no horizontal only a vertical oblique slant

The rational function R(x)=(x^3-8)/x has a/an [Select ] " asymptote. Choose the best answer. [Select] no horizontal only a vertical oblique slant
Transcript text: The rational function $\mathbb{R}(x)=\frac{x^{3}-8}{x}$ has $\mathrm{a} / \mathrm{an}$ [Select ] " asymptote. Choose the best answer. [Select] no horizontal only a vertical oblique slant
failed

Solution

failed
failed

Solution Steps

To determine the type of asymptote for the rational function \(\mathbb{R}(x) = \frac{x^3 - 8}{x}\), we need to analyze the degrees of the polynomial in the numerator and the denominator. The degree of the numerator is 3, and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator by exactly one, the function has an oblique (or slant) asymptote. We can find the equation of the oblique asymptote by performing polynomial long division on the numerator by the denominator.

Step 1: Identify the Function

We are given the rational function

\[ \mathbb{R}(x) = \frac{x^3 - 8}{x}. \]

Step 2: Determine the Degrees

The degree of the numerator \(x^3 - 8\) is 3, and the degree of the denominator \(x\) is 1. Since the degree of the numerator is greater than the degree of the denominator by exactly one, we expect an oblique (or slant) asymptote.

Step 3: Perform Polynomial Long Division

To find the oblique asymptote, we perform polynomial long division of \(x^3 - 8\) by \(x\):

  1. Divide the leading term \(x^3\) by \(x\) to get \(x^2\).
  2. Multiply \(x^2\) by \(x\) to get \(x^3\).
  3. Subtract \(x^3\) from \(x^3 - 8\) to get \(-8\).
  4. The remainder is \(-8\), which does not affect the asymptote.

Thus, the quotient from the division is

\[ x^2. \]

Final Answer

The oblique asymptote of the function \(\mathbb{R}(x)\) is

\[ \boxed{y = x^2}. \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Two cars start moving from the same point. One travels south at 20 mi / h and the other travels west at 15 mi / h. At what rate (in mi/h) is the distance between the cars increasing three hours later?
failedAnswered
A car dealership decreased the price of a certain car by 5%. The original price was 49,800. (a) Fill in the blank to write the new price in terms of the original price. Write your answer as a decimal. New price = (blank) × Original price (b) Use your answer in part (a) to determine the new price. New price: (blank) (blank)
failedAnswered
Which treatment is considered a last resort for treating arthritis? Select one: weight loss and exercise, total joint replacement, MACl - matrix associated cartilage implantation, microfracture
failedAnswered
(0,-3), m=3/5 1 of 2: Find the equation of the line for the y-intercept and slope given above.
failedAnswered
Adding an Internet channel is attractive to retailers with strong brand names but limited locations and distribution. makes it difficult for retailers to develop extensive purchase histories of their customers. allows customers the opportunity to use all five senses when examining and evaluating products. reduces the number of product alternatives that can be offered to consumers.