Rational Functions: Asymptotes and Discontinuities
Master the key concepts of rational functions including asymptotes, holes, and domain restrictions.
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Questions Covered in This Set
8 cards to master
What is a rational function?
A function that can be written as the ratio of two polynomials: f(x) = P(x)/Q(x), where Q(x) ≠ 0
What is a removable discontinuity (hole)?
A point where a common factor cancels from numerator and denominator, leaving the function undefined at that x-value but approaching a finite y-value
How do you find vertical asymptotes?
Factor numerator and denominator, cancel common factors, then set remaining denominator equal to zero and solve
What is the horizontal asymptote when degree of numerator < degree of denominator?
y = 0
What is the horizontal asymptote when degree of numerator = degree of denominator?
y = (leading coefficient of numerator)/(leading coefficient of denominator)
When does an oblique (slant) asymptote occur?
When the degree of the numerator is exactly one more than the degree of the denominator (n = m + 1)
How do you find an oblique asymptote?
Perform polynomial long division and use the quotient (ignoring the remainder) as the equation of the oblique asymptote
What happens as x approaches a vertical asymptote?
The function values grow without bound, approaching positive or negative infinity
