Math Problem Statement

Give the equations of any​ vertical, horizontal, or oblique asymptotes for the graph of the rational function. f left parenthesis x right parenthesis equals StartFraction 5 minus 5 x Over 5 x plus 3 EndFraction Question content area bottom Part 1 Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The equation of the vertical asymptote is xequals    enter your response here. ​(Type an integer or a fraction. Simplify your​ answer.) B. There is no vertical asymptote. Part 2 Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The equation of the horizontal asymptote is yequals    enter your response here. ​(Type an integer or a fraction. Simplify your​ answer.) B. There is no horizontal asymptote. Part 3 Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The equation of the oblique asymptote is yequals    enter your response here. ​(Type an integer or a fraction. Simplify your​ answer.) B. There is no oblique asymptote.

Solution

To find the asymptotes for the rational function f(x)=5−5x5x+3f(x) = \frac{5 - 5x}{5x + 3}, we will examine vertical, horizontal, and oblique asymptotes.

Part 1: Vertical Asymptote

Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at the same point).

  1. Set the denominator equal to zero: 5x+3=05x + 3 = 0 Solving for xx: 5x=−3  ⟹  x=−355x = -3 \implies x = -\frac{3}{5}

Answer: A. The equation of the vertical asymptote is x=−35x = -\frac{3}{5}.

Part 2: Horizontal Asymptote

To find horizontal asymptotes, we compare the degrees of the polynomial in the numerator and denominator.

  • The degree of the numerator (5 - 5x) is 1 (linear).
  • The degree of the denominator (5x + 3) is also 1 (linear).

When the degrees are the same, the horizontal asymptote is given by the ratio of the leading coefficients: Horizontal Asymptote: y=−55=−1\text{Horizontal Asymptote: } y = \frac{-5}{5} = -1

Answer: A. The equation of the horizontal asymptote is y=−1y = -1.

Part 3: Oblique Asymptote

Oblique (or slant) asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. Since both degrees are equal here, there is no oblique asymptote.

Answer: B. There is no oblique asymptote.

Summary of Answers:

  1. Vertical Asymptote: x=−35x = -\frac{3}{5}
  2. Horizontal Asymptote: y=−1y = -1
  3. Oblique Asymptote: None

Do you want any more details or have any questions? Here are some related questions you might find interesting:

  1. How do you determine the degree of a polynomial?
  2. What is the significance of asymptotes in graphing rational functions?
  3. How can you sketch the graph of the function given its asymptotes?
  4. What happens to the function as xx approaches the vertical asymptote?
  5. Can a rational function have more than one vertical asymptote?

Tip: Always check both the numerator and denominator to ensure there are no common factors that could affect the asymptotes.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Asymptotes
Algebra

Formulas

Vertical Asymptote: set denominator to zero
Horizontal Asymptote: leading coefficients ratio

Theorems

Vertical and horizontal asymptote rules

Suitable Grade Level

Grades 9-11

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