Rational Functions and Vertical Asymptotes - AP Precalculus
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Does $f(x) = \frac{x + 2}{x^2 + 1}$ have a vertical asymptote?
Does $f(x) = \frac{x + 2}{x^2 + 1}$ have a vertical asymptote?
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No vertical asymptotes. Denominator $x^2 + 1$ has no real zeros since it's always positive.
No vertical asymptotes. Denominator $x^2 + 1$ has no real zeros since it's always positive.
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Does $f(x) = \frac{1}{x^2 + 4}$ have a vertical asymptote?
Does $f(x) = \frac{1}{x^2 + 4}$ have a vertical asymptote?
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No vertical asymptotes. Denominator $x^2 + 4$ has no real zeros since discriminant is negative.
No vertical asymptotes. Denominator $x^2 + 4$ has no real zeros since discriminant is negative.
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What is the vertical asymptote of $f(x) = \frac{1}{x^2 - 9}$?
What is the vertical asymptote of $f(x) = \frac{1}{x^2 - 9}$?
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Vertical asymptotes at $x = 3$ and $x = -3$. Factor denominator as $(x-3)(x+3)$ and set equal to zero.
Vertical asymptotes at $x = 3$ and $x = -3$. Factor denominator as $(x-3)(x+3)$ and set equal to zero.
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Find the vertical asymptote for $f(x) = \frac{4}{x^2 - 4x + 3}$.
Find the vertical asymptote for $f(x) = \frac{4}{x^2 - 4x + 3}$.
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Vertical asymptotes at $x = 1$ and $x = 3$. Factor denominator as $(x-1)(x-3)$ and solve for zeros.
Vertical asymptotes at $x = 1$ and $x = 3$. Factor denominator as $(x-1)(x-3)$ and solve for zeros.
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State the vertical asymptote of $f(x) = \frac{x}{x^2 - 16}$.
State the vertical asymptote of $f(x) = \frac{x}{x^2 - 16}$.
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Vertical asymptotes at $x = 4$ and $x = -4$. Factor denominator as $(x-4)(x+4)$ and solve for zeros.
Vertical asymptotes at $x = 4$ and $x = -4$. Factor denominator as $(x-4)(x+4)$ and solve for zeros.
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Identify the vertical asymptote for $f(x) = \frac{2x - 1}{x^2 - 4x}$.
Identify the vertical asymptote for $f(x) = \frac{2x - 1}{x^2 - 4x}$.
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Vertical asymptotes at $x = 0$ and $x = 4$. Factor denominator as $x(x-4)$ and set each factor to zero.
Vertical asymptotes at $x = 0$ and $x = 4$. Factor denominator as $x(x-4)$ and set each factor to zero.
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What is the definition of a rational function?
What is the definition of a rational function?
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A function of the form $f(x) = \frac{P(x)}{Q(x)}$ where $P$ and $Q$ are polynomials. Two polynomials create a fraction with specific asymptotic behavior.
A function of the form $f(x) = \frac{P(x)}{Q(x)}$ where $P$ and $Q$ are polynomials. Two polynomials create a fraction with specific asymptotic behavior.
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What condition creates a vertical asymptote in a rational function?
What condition creates a vertical asymptote in a rational function?
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A vertical asymptote occurs where $Q(x) = 0$ and $P(x) \neq 0$. Denominator zero with nonzero numerator creates infinite discontinuity.
A vertical asymptote occurs where $Q(x) = 0$ and $P(x) \neq 0$. Denominator zero with nonzero numerator creates infinite discontinuity.
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Identify the vertical asymptote of $f(x) = \frac{1}{x - 3}$.
Identify the vertical asymptote of $f(x) = \frac{1}{x - 3}$.
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Vertical asymptote at $x = 3$. Set denominator $x - 3 = 0$ to find where function is undefined.
Vertical asymptote at $x = 3$. Set denominator $x - 3 = 0$ to find where function is undefined.
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State the vertical asymptote of $f(x) = \frac{x^2}{x^2 - 4}$.
State the vertical asymptote of $f(x) = \frac{x^2}{x^2 - 4}$.
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Vertical asymptotes at $x = 2$ and $x = -2$. Factor denominator as $(x-2)(x+2)$ and set each factor to zero.
Vertical asymptotes at $x = 2$ and $x = -2$. Factor denominator as $(x-2)(x+2)$ and set each factor to zero.
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What is the vertical asymptote of $f(x) = \frac{x + 1}{x^2 - x - 6}$?
What is the vertical asymptote of $f(x) = \frac{x + 1}{x^2 - x - 6}$?
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Vertical asymptotes at $x = 3$ and $x = -2$. Factor denominator $x^2 - x - 6 = (x-3)(x+2)$ and solve.
Vertical asymptotes at $x = 3$ and $x = -2$. Factor denominator $x^2 - x - 6 = (x-3)(x+2)$ and solve.
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Find the vertical asymptote for $f(x) = \frac{2x}{x^2 - 9}$.
Find the vertical asymptote for $f(x) = \frac{2x}{x^2 - 9}$.
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Vertical asymptotes at $x = 3$ and $x = -3$. Factor denominator as $(x-3)(x+3)$ and set equal to zero.
Vertical asymptotes at $x = 3$ and $x = -3$. Factor denominator as $(x-3)(x+3)$ and set equal to zero.
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What is the vertical asymptote of $f(x) = \frac{x + 1}{x^2 - x - 6}$?
What is the vertical asymptote of $f(x) = \frac{x + 1}{x^2 - x - 6}$?
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Vertical asymptotes at $x = 3$ and $x = -2$. Factor denominator $x^2 - x - 6 = (x-3)(x+2)$ and solve.
Vertical asymptotes at $x = 3$ and $x = -2$. Factor denominator $x^2 - x - 6 = (x-3)(x+2)$ and solve.
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Determine the vertical asymptote for $f(x) = \frac{x + 2}{x^2 - 2x - 3}$.
Determine the vertical asymptote for $f(x) = \frac{x + 2}{x^2 - 2x - 3}$.
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Vertical asymptotes at $x = 3$ and $x = -1$. Factor denominator as $(x-3)(x+1)$ and solve for zeros.
Vertical asymptotes at $x = 3$ and $x = -1$. Factor denominator as $(x-3)(x+1)$ and solve for zeros.
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State the vertical asymptote of $f(x) = \frac{x^2 + 1}{x^2 + x - 2}$.
State the vertical asymptote of $f(x) = \frac{x^2 + 1}{x^2 + x - 2}$.
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Vertical asymptotes at $x = 1$ and $x = -2$. Factor denominator as $(x-1)(x+2)$ and solve for zeros.
Vertical asymptotes at $x = 1$ and $x = -2$. Factor denominator as $(x-1)(x+2)$ and solve for zeros.
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Find the vertical asymptote for $f(x) = \frac{2x + 1}{x^2 - 5x + 6}$.
Find the vertical asymptote for $f(x) = \frac{2x + 1}{x^2 - 5x + 6}$.
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Vertical asymptotes at $x = 2$ and $x = 3$. Factor denominator as $(x-2)(x-3)$ and solve for zeros.
Vertical asymptotes at $x = 2$ and $x = 3$. Factor denominator as $(x-2)(x-3)$ and solve for zeros.
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Identify the vertical asymptote for $f(x) = \frac{x - 4}{x^3 - 8}$.
Identify the vertical asymptote for $f(x) = \frac{x - 4}{x^3 - 8}$.
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Vertical asymptote at $x = 2$. Factor $x^3 - 8 = (x-2)(x^2+2x+4)$; only real zero is $x=2$.
Vertical asymptote at $x = 2$. Factor $x^3 - 8 = (x-2)(x^2+2x+4)$; only real zero is $x=2$.
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What is the vertical asymptote of $f(x) = \frac{2}{x^2 - 4}$?
What is the vertical asymptote of $f(x) = \frac{2}{x^2 - 4}$?
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Vertical asymptotes at $x = 2$ and $x = -2$. Factor denominator as $(x-2)(x+2)$ and solve for zeros.
Vertical asymptotes at $x = 2$ and $x = -2$. Factor denominator as $(x-2)(x+2)$ and solve for zeros.
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State the vertical asymptote for $f(x) = \frac{x^2 - 1}{x - 1}$.
State the vertical asymptote for $f(x) = \frac{x^2 - 1}{x - 1}$.
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Removable discontinuity at $x = 1$. Common factor $ (x-1) $ cancels, leaving a hole instead of asymptote.
Removable discontinuity at $x = 1$. Common factor $ (x-1) $ cancels, leaving a hole instead of asymptote.
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What is the vertical asymptote of $f(x) = \frac{5}{x^2 + 2x}$?
What is the vertical asymptote of $f(x) = \frac{5}{x^2 + 2x}$?
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Vertical asymptotes at $x = 0$ and $x = -2$. Factor denominator as $x(x+2)$ and solve for zeros.
Vertical asymptotes at $x = 0$ and $x = -2$. Factor denominator as $x(x+2)$ and solve for zeros.
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Identify the vertical asymptote for $f(x) = \frac{x - 4}{x^3 - 8}$.
Identify the vertical asymptote for $f(x) = \frac{x - 4}{x^3 - 8}$.
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Vertical asymptote at $x = 2$. Factor $x^3 - 8 = (x-2)(x^2+2x+4)$; only real zero is $x=2$.
Vertical asymptote at $x = 2$. Factor $x^3 - 8 = (x-2)(x^2+2x+4)$; only real zero is $x=2$.
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Determine the vertical asymptote for $f(x) = \frac{x + 2}{x^2 - 2x - 3}$.
Determine the vertical asymptote for $f(x) = \frac{x + 2}{x^2 - 2x - 3}$.
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Vertical asymptotes at $x = 3$ and $x = -1$. Factor denominator as $(x-3)(x+1)$ and solve for zeros.
Vertical asymptotes at $x = 3$ and $x = -1$. Factor denominator as $(x-3)(x+1)$ and solve for zeros.
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Identify the vertical asymptote for $f(x) = \frac{x^2 - 9}{x - 3}$.
Identify the vertical asymptote for $f(x) = \frac{x^2 - 9}{x - 3}$.
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Removable discontinuity at $x = 3$. Common factor $ (x-3) $ cancels, creating a hole at $x=3$.
Removable discontinuity at $x = 3$. Common factor $ (x-3) $ cancels, creating a hole at $x=3$.
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What is the vertical asymptote of $f(x) = \frac{1}{x^2 + 2x + 1}$?
What is the vertical asymptote of $f(x) = \frac{1}{x^2 + 2x + 1}$?
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Vertical asymptote at $x = -1$. Factor denominator as $(x+1)^2$; repeated zero at $x=-1$.
Vertical asymptote at $x = -1$. Factor denominator as $(x+1)^2$; repeated zero at $x=-1$.
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State the vertical asymptote for $f(x) = \frac{x^2 - 1}{x - 1}$.
State the vertical asymptote for $f(x) = \frac{x^2 - 1}{x - 1}$.
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Removable discontinuity at $x = 1$. Common factor $(x-1)$ cancels, leaving a hole instead of asymptote.
Removable discontinuity at $x = 1$. Common factor $(x-1)$ cancels, leaving a hole instead of asymptote.
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What is the vertical asymptote of $f(x) = \frac{x}{x^3 - 27}$?
What is the vertical asymptote of $f(x) = \frac{x}{x^3 - 27}$?
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Vertical asymptote at $x = 3$. Factor $x^3 - 27 = (x-3)(x^2+3x+9)$; only real zero is $x=3$.
Vertical asymptote at $x = 3$. Factor $x^3 - 27 = (x-3)(x^2+3x+9)$; only real zero is $x=3$.
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Find the vertical asymptote for $f(x) = \frac{2x + 1}{x^2 - 5x + 6}$.
Find the vertical asymptote for $f(x) = \frac{2x + 1}{x^2 - 5x + 6}$.
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Vertical asymptotes at $x = 2$ and $x = 3$. Factor denominator as $(x-2)(x-3)$ and solve for zeros.
Vertical asymptotes at $x = 2$ and $x = 3$. Factor denominator as $(x-2)(x-3)$ and solve for zeros.
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What is the vertical asymptote of $f(x) = \frac{1}{x^2 - 25}$?
What is the vertical asymptote of $f(x) = \frac{1}{x^2 - 25}$?
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Vertical asymptotes at $x = 5$ and $x = -5$. Factor denominator as $(x-5)(x+5)$ and solve for zeros.
Vertical asymptotes at $x = 5$ and $x = -5$. Factor denominator as $(x-5)(x+5)$ and solve for zeros.
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State the vertical asymptote of $f(x) = \frac{3}{x^2 - 9x}$.
State the vertical asymptote of $f(x) = \frac{3}{x^2 - 9x}$.
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Vertical asymptotes at $x = 0$ and $x = 9$. Factor denominator as $x(x-9)$ and set each factor to zero.
Vertical asymptotes at $x = 0$ and $x = 9$. Factor denominator as $x(x-9)$ and set each factor to zero.
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What is the vertical asymptote of $f(x) = \frac{2x}{x^2 - 1}$?
What is the vertical asymptote of $f(x) = \frac{2x}{x^2 - 1}$?
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Vertical asymptotes at $x = 1$ and $x = -1$. Factor denominator as $(x-1)(x+1)$ and solve for zeros.
Vertical asymptotes at $x = 1$ and $x = -1$. Factor denominator as $(x-1)(x+1)$ and solve for zeros.
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