Math Problem Statement

Find the equations of any vertical asymptotes for the function below. ​f(x)equalsStartFraction x squared minus 3 x minus 10 Over x squared minus 5 x minus 14 EndFraction

Question content area bottom Part 1 Find the vertical​ asymptote(s). Select the correct choice below​ and, if​ necessary, fill in the answer​ box(es) to complete your choice. A. The function has two vertical asymptotes. The leftmost asymptote is    enter your response here and the rightmost asymptote is    enter your response here. ​(Type equations.) B. The function has three vertical asymptotes. The asymptotes in order from leftmost to rightmost are    enter your response here​,    enter your response here​, and    enter your response here. ​(Type equations.) C. The function has four vertical asymptotes. The asymptotes in order from leftmost to rightmost are    enter your response here​,    enter your response here​,    enter your response here​, and    enter your response here. ​(Type equations.) D. The function has one vertical​ asymptote,    enter your response here. ​(Type an​ equation.) E. The function has no vertical asymptotes.

Solution

The given function is:

f(x)=x23x10x25x14f(x) = \frac{x^2 - 3x - 10}{x^2 - 5x - 14}

Steps to find the vertical asymptotes:

  1. Vertical asymptotes occur when the denominator of a rational function equals zero, provided the numerator doesn't also equal zero at the same points (which would indicate a hole instead of an asymptote).

  2. First, we factor both the numerator and the denominator.

Numerator: x23x10x^2 - 3x - 10

Factorizing:

x23x10=(x5)(x+2)x^2 - 3x - 10 = (x - 5)(x + 2)

Denominator: x25x14x^2 - 5x - 14

Factorizing:

x25x14=(x7)(x+2)x^2 - 5x - 14 = (x - 7)(x + 2)

  1. Identify potential vertical asymptotes: The vertical asymptotes occur where the denominator is zero but not the numerator.

Setting the denominator equal to zero:

x7=0orx+2=0x - 7 = 0 \quad \text{or} \quad x + 2 = 0

x=7orx=2x = 7 \quad \text{or} \quad x = -2

  1. Check for holes: If the numerator and denominator share a common factor, a hole will occur at that value of xx. In this case, both the numerator and the denominator have a factor of (x+2)(x + 2), meaning there is a hole at x=2x = -2, not a vertical asymptote.

Conclusion:

The only vertical asymptote occurs at x=7x = 7.

Thus, the correct answer is:

  • D. The function has one vertical asymptote, x=7x = 7.

Would you like any further details or have any questions?

Here are 5 related questions for practice:

  1. How do you find horizontal asymptotes for rational functions?
  2. What is the difference between a hole and a vertical asymptote?
  3. Can a function have both a vertical asymptote and a hole at the same point?
  4. How do you factor quadratic polynomials?
  5. What happens to the function as it approaches its vertical asymptote?

Tip: Always simplify rational functions by canceling out common factors between the numerator and denominator before determining vertical asymptotes or holes.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Rational Functions
Vertical Asymptotes
Factoring Quadratics
Holes in Functions

Formulas

Vertical asymptotes occur where the denominator equals zero and the numerator is non-zero.

Theorems

Zero of the denominator theorem for asymptotes
Factoring quadratics

Suitable Grade Level

Grades 10-12