Rational Functions and End Behavior - AP Precalculus
Card 1 of 30
What does the end behavior of $f(x) = \frac{x^3}{x+1}$ as $x \to \infty$ approach?
What does the end behavior of $f(x) = \frac{x^3}{x+1}$ as $x \to \infty$ approach?
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$f(x)$ approaches $\infty$ as $x \to \infty$. Numerator degree exceeds denominator degree.
$f(x)$ approaches $\infty$ as $x \to \infty$. Numerator degree exceeds denominator degree.
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Determine the horizontal asymptote of $f(x) = \frac{x+2}{x^2+3x+1}$.
Determine the horizontal asymptote of $f(x) = \frac{x+2}{x^2+3x+1}$.
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Horizontal asymptote at $y = 0$. Numerator degree less than denominator degree.
Horizontal asymptote at $y = 0$. Numerator degree less than denominator degree.
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What is the end behavior of $f(x) = \frac{2x^3}{x^2+1}$ as $x \to -\infty$?
What is the end behavior of $f(x) = \frac{2x^3}{x^2+1}$ as $x \to -\infty$?
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$f(x)$ approaches $-\infty$ as $x \to -\infty$. Numerator degree exceeds denominator, goes to $-\infty$.
$f(x)$ approaches $-\infty$ as $x \to -\infty$. Numerator degree exceeds denominator, goes to $-\infty$.
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Determine the vertical asymptote of $f(x) = \frac{x^2 + 1}{x^2 - 4x + 3}$.
Determine the vertical asymptote of $f(x) = \frac{x^2 + 1}{x^2 - 4x + 3}$.
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Vertical asymptotes at $x = 3$ and $x = 1$. Factor denominator: $(x-3)(x-1) = 0$.
Vertical asymptotes at $x = 3$ and $x = 1$. Factor denominator: $(x-3)(x-1) = 0$.
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Determine the horizontal asymptote of $f(x) = \frac{x+2}{x^2+3x+1}$.
Determine the horizontal asymptote of $f(x) = \frac{x+2}{x^2+3x+1}$.
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Horizontal asymptote at $y = 0$. Numerator degree less than denominator degree.
Horizontal asymptote at $y = 0$. Numerator degree less than denominator degree.
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Identify the removable discontinuity of $f(x) = \frac{(x-2)(x+3)}{x-2}$.
Identify the removable discontinuity of $f(x) = \frac{(x-2)(x+3)}{x-2}$.
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Removable discontinuity at $x = 2$. Common factor $(x-2)$ cancels out.
Removable discontinuity at $x = 2$. Common factor $(x-2)$ cancels out.
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Identify the horizontal asymptote for $f(x) = \frac{2x^2}{x^2 + 1}$.
Identify the horizontal asymptote for $f(x) = \frac{2x^2}{x^2 + 1}$.
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Horizontal asymptote at $y = 2$. Same degree: ratio of leading coefficients $\frac{2}{1}$.
Horizontal asymptote at $y = 2$. Same degree: ratio of leading coefficients $\frac{2}{1}$.
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What is the vertical asymptote of $f(x) = \frac{2x^2}{x^2 - 9}$?
What is the vertical asymptote of $f(x) = \frac{2x^2}{x^2 - 9}$?
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Vertical asymptotes at $x = 3$ and $x = -3$. Factor denominator: $(x-3)(x+3) = 0$.
Vertical asymptotes at $x = 3$ and $x = -3$. Factor denominator: $(x-3)(x+3) = 0$.
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State the horizontal asymptote for $f(x) = \frac{4x+3}{7x+8}$.
State the horizontal asymptote for $f(x) = \frac{4x+3}{7x+8}$.
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Horizontal asymptote at $y = \frac{4}{7}$. Same degree: ratio of leading coefficients $\frac{4}{7}$.
Horizontal asymptote at $y = \frac{4}{7}$. Same degree: ratio of leading coefficients $\frac{4}{7}$.
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What is the end behavior of $f(x) = \frac{3x^4}{x^2 + 1}$ as $x \to \infty$?
What is the end behavior of $f(x) = \frac{3x^4}{x^2 + 1}$ as $x \to \infty$?
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$f(x)$ approaches $\infty$ as $x \to \infty$. Numerator degree exceeds denominator degree.
$f(x)$ approaches $\infty$ as $x \to \infty$. Numerator degree exceeds denominator degree.
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Does $f(x) = \frac{x^2-4}{x^2+4}$ have a horizontal asymptote?
Does $f(x) = \frac{x^2-4}{x^2+4}$ have a horizontal asymptote?
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Yes, $y = 1$. Same degree polynomials have horizontal asymptote.
Yes, $y = 1$. Same degree polynomials have horizontal asymptote.
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Find the vertical asymptote of $f(x) = \frac{x^2 + 1}{x - 5}$.
Find the vertical asymptote of $f(x) = \frac{x^2 + 1}{x - 5}$.
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Vertical asymptote at $x = 5$. Set denominator equal to zero: $x - 5 = 0$.
Vertical asymptote at $x = 5$. Set denominator equal to zero: $x - 5 = 0$.
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What is the horizontal asymptote for $f(x) = \frac{5x+1}{2x-3}$?
What is the horizontal asymptote for $f(x) = \frac{5x+1}{2x-3}$?
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Horizontal asymptote at $y = \frac{5}{2}$. Same degree: ratio of leading coefficients $\frac{5}{2}$.
Horizontal asymptote at $y = \frac{5}{2}$. Same degree: ratio of leading coefficients $\frac{5}{2}$.
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What is the domain of $f(x) = \frac{2}{x-4}$?
What is the domain of $f(x) = \frac{2}{x-4}$?
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All real numbers except $x = 4$. Exclude values where denominator equals zero.
All real numbers except $x = 4$. Exclude values where denominator equals zero.
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State the vertical asymptotes for $f(x) = \frac{1}{x^2 - 4x + 4}$.
State the vertical asymptotes for $f(x) = \frac{1}{x^2 - 4x + 4}$.
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Vertical asymptote at $x = 2$. Perfect square: $(x-2)^2 = 0$ gives $x = 2$.
Vertical asymptote at $x = 2$. Perfect square: $(x-2)^2 = 0$ gives $x = 2$.
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What is the end behavior of $f(x) = \frac{2x^3}{x^2+1}$ as $x \to -\infty$?
What is the end behavior of $f(x) = \frac{2x^3}{x^2+1}$ as $x \to -\infty$?
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$f(x)$ approaches $-\infty$ as $x \to -\infty$. Numerator degree exceeds denominator, goes to $-\infty$.
$f(x)$ approaches $-\infty$ as $x \to -\infty$. Numerator degree exceeds denominator, goes to $-\infty$.
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Identify the horizontal asymptote for $f(x) = \frac{5x^3}{3x^3 + 1}$.
Identify the horizontal asymptote for $f(x) = \frac{5x^3}{3x^3 + 1}$.
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Horizontal asymptote at $y = \frac{5}{3}$. Same degree: ratio of leading coefficients $\frac{5}{3}$.
Horizontal asymptote at $y = \frac{5}{3}$. Same degree: ratio of leading coefficients $\frac{5}{3}$.
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What is the domain of $f(x) = \frac{1}{x^2 - 9}$?
What is the domain of $f(x) = \frac{1}{x^2 - 9}$?
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All real numbers except $x = 3$ and $x = -3$. Exclude values where $x^2 - 9 = 0$.
All real numbers except $x = 3$ and $x = -3$. Exclude values where $x^2 - 9 = 0$.
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What is the end behavior of $f(x) = \frac{x^2 + 2x + 1}{x^2}$ as $x \to \infty$?
What is the end behavior of $f(x) = \frac{x^2 + 2x + 1}{x^2}$ as $x \to \infty$?
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$f(x)$ approaches $1$ as $x \to \infty$. Expand and simplify: $\frac{x^2+2x+1}{x^2} = 1 + \frac{2}{x} + \frac{1}{x^2}$.
$f(x)$ approaches $1$ as $x \to \infty$. Expand and simplify: $\frac{x^2+2x+1}{x^2} = 1 + \frac{2}{x} + \frac{1}{x^2}$.
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State the vertical asymptotes for $f(x) = \frac{x^2-4}{x^2-5x+6}$.
State the vertical asymptotes for $f(x) = \frac{x^2-4}{x^2-5x+6}$.
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Vertical asymptotes at $x = 3$ and $x = 2$. Factor denominator: $(x-3)(x-2) = 0$.
Vertical asymptotes at $x = 3$ and $x = 2$. Factor denominator: $(x-3)(x-2) = 0$.
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Identify the removable discontinuity of $f(x) = \frac{x^2-1}{x^2+x}$.
Identify the removable discontinuity of $f(x) = \frac{x^2-1}{x^2+x}$.
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Removable discontinuity at $x = -1$. Factor: $\frac{(x-1)(x+1)}{x(x+1)}$ cancels $(x+1)$.
Removable discontinuity at $x = -1$. Factor: $\frac{(x-1)(x+1)}{x(x+1)}$ cancels $(x+1)$.
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What is the domain of $f(x) = \frac{x^2 + 3x - 4}{x^2 - 4}$?
What is the domain of $f(x) = \frac{x^2 + 3x - 4}{x^2 - 4}$?
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All real numbers except $x = 2$ and $x = -2$. Exclude values where $x^2 - 4 = 0$.
All real numbers except $x = 2$ and $x = -2$. Exclude values where $x^2 - 4 = 0$.
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What is the horizontal asymptote for $f(x) = \frac{2x}{x^2+3}$?
What is the horizontal asymptote for $f(x) = \frac{2x}{x^2+3}$?
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Horizontal asymptote at $y = 0$. Numerator degree less than denominator degree.
Horizontal asymptote at $y = 0$. Numerator degree less than denominator degree.
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Determine the vertical asymptote of $f(x) = \frac{x^2 + 1}{x^2 - 4x + 3}$.
Determine the vertical asymptote of $f(x) = \frac{x^2 + 1}{x^2 - 4x + 3}$.
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Vertical asymptotes at $x = 3$ and $x = 1$. Factor denominator: $(x-3)(x-1) = 0$.
Vertical asymptotes at $x = 3$ and $x = 1$. Factor denominator: $(x-3)(x-1) = 0$.
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Determine the end behavior of $f(x) = \frac{x^2}{x^3+1}$ as $x \to -\infty$.
Determine the end behavior of $f(x) = \frac{x^2}{x^3+1}$ as $x \to -\infty$.
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$f(x)$ approaches $0$ as $x \to -\infty$. Degree of numerator less than denominator.
$f(x)$ approaches $0$ as $x \to -\infty$. Degree of numerator less than denominator.
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What is the end behavior of $f(x) = \frac{2x^2}{x^3 + 1}$ as $x \to \infty$?
What is the end behavior of $f(x) = \frac{2x^2}{x^3 + 1}$ as $x \to \infty$?
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$f(x)$ approaches $0$ as $x \to \infty$. Numerator degree less than denominator degree.
$f(x)$ approaches $0$ as $x \to \infty$. Numerator degree less than denominator degree.
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What is the end behavior of $f(x) = \frac{3x^3}{2x^3 + 5}$ as $x$ approaches infinity?
What is the end behavior of $f(x) = \frac{3x^3}{2x^3 + 5}$ as $x$ approaches infinity?
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$f(x)$ approaches $\frac{3}{2}$ as $x \to \infty$. Same degree: divide leading coefficients $\frac{3}{2}$.
$f(x)$ approaches $\frac{3}{2}$ as $x \to \infty$. Same degree: divide leading coefficients $\frac{3}{2}$.
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State the horizontal asymptote of $f(x) = \frac{2x^2+3}{x^2-1}$.
State the horizontal asymptote of $f(x) = \frac{2x^2+3}{x^2-1}$.
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Horizontal asymptote at $y = 2$. Same degree polynomials: ratio of leading coefficients.
Horizontal asymptote at $y = 2$. Same degree polynomials: ratio of leading coefficients.
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Identify the vertical asymptote for $f(x) = \frac{x+1}{x-3}$.
Identify the vertical asymptote for $f(x) = \frac{x+1}{x-3}$.
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Vertical asymptote at $x = 3$. Set denominator equal to zero: $x - 3 = 0$.
Vertical asymptote at $x = 3$. Set denominator equal to zero: $x - 3 = 0$.
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What is a rational function?
What is a rational function?
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A function of the form $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials. The numerator and denominator must be polynomials.
A function of the form $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials. The numerator and denominator must be polynomials.
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