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:
Steps to find the vertical asymptotes:
-
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).
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First, we factor both the numerator and the denominator.
Numerator:
Factorizing:
Denominator:
Factorizing:
- Identify potential vertical asymptotes: The vertical asymptotes occur where the denominator is zero but not the numerator.
Setting the denominator equal to zero:
- Check for holes: If the numerator and denominator share a common factor, a hole will occur at that value of . In this case, both the numerator and the denominator have a factor of , meaning there is a hole at , not a vertical asymptote.
Conclusion:
The only vertical asymptote occurs at .
Thus, the correct answer is:
- D. The function has one vertical asymptote, .
Would you like any further details or have any questions?
Here are 5 related questions for practice:
- How do you find horizontal asymptotes for rational functions?
- What is the difference between a hole and a vertical asymptote?
- Can a function have both a vertical asymptote and a hole at the same point?
- How do you factor quadratic polynomials?
- 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.
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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
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