AP Precalculus Flashcards: Rational Functions And End Behavior

Study Rational Functions And End Behavior in AP Precalculus with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Precalculus

Rational Functions And End Behavior

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QUESTION

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State the vertical asymptotes for $f(x) = \frac{1}{x^2 - 4x + 4}$ .

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ANSWER

Vertical asymptote at $x = 2$ . Perfect square: $(x-2)^2 = 0$ gives $x = 2$ .

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What this deck covers

This deck focuses on Rational Functions And End Behavior, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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Flashcard 1: State the vertical asymptotes for $f(x) = \frac{1}{x^2 - 4x + 4}$ .

Answer: Vertical asymptote at $x = 2$ . Perfect square: $(x-2)^2 = 0$ gives $x = 2$ .

Flashcard 2: What is the end behavior of $f(x) = \frac{3x^3}{2x^3 + 5}$ as $x$ approaches infinity?

Answer: $f(x)$ approaches $\frac{3}{2}$ as $x \to \infty$ . Same degree: divide leading coefficients $\frac{3}{2}$ .

Flashcard 3: State the vertical asymptotes for $f(x) = \frac{x^2-1}{x^2-4}$ .

Answer: Vertical asymptotes at $x = 2$ and $x = -2$ . Factor denominator: $(x-2)(x+2) = 0$ .

Flashcard 4: What is a removable discontinuity in rational functions?

Answer: A hole where $p(x)$ and $q(x)$ share a common factor. Common factors cancel, creating a hole.

Flashcard 5: What is the end behavior of $f(x) = \frac{2x^2}{x^3 + 1}$ as $x \to \infty$ ?

Answer: $f(x)$ approaches $0$ as $x \to \infty$ . Numerator degree less than denominator degree.

Flashcard 6: State the horizontal asymptote of $f(x) = \frac{2x^2+3}{x^2-1}$ .

Answer: Horizontal asymptote at $y = 2$ . Same degree polynomials: ratio of leading coefficients.

Flashcard 7: What is the horizontal asymptote of $f(x) = \frac{3x}{x + 4}$ ?

Answer: Horizontal asymptote at $y = 3$ . Same degree: ratio of leading coefficients $\frac{3}{1}$ .

Flashcard 8: Determine the vertical asymptote of $f(x) = \frac{x^2 + 1}{x^2 - 4x + 3}$ .

Answer: Vertical asymptotes at $x = 3$ and $x = 1$ . Factor denominator: $(x-3)(x-1) = 0$ .

Flashcard 9: What is the horizontal asymptote for $f(x) = \frac{2x}{x^2+3}$ ?

Answer: Horizontal asymptote at $y = 0$ . Numerator degree less than denominator degree.

Flashcard 10: What is the end behavior of $f(x) = \frac{x^2 + 2x + 1}{x^2}$ as $x \to \infty$ ?

Answer: $f(x)$ approaches $1$ as $x \to \infty$ . Expand and simplify: $\frac{x^2+2x+1}{x^2} = 1 + \frac{2}{x} + \frac{1}{x^2}$ .

Flashcard 11: What is the end behavior of $f(x) = \frac{2x^3}{x^2+1}$ as $x \to -\infty$ ?

Answer: $f(x)$ approaches $-\infty$ as $x \to -\infty$ . Numerator degree exceeds denominator, goes to $-\infty$ .

Flashcard 12: Find the vertical asymptote of $f(x) = \frac{x^2 + 1}{x - 5}$ .

Answer: Vertical asymptote at $x = 5$ . Set denominator equal to zero: $x - 5 = 0$ .

Flashcard 13: State the horizontal asymptote for $f(x) = \frac{4x+3}{7x+8}$ .

Answer: Horizontal asymptote at $y = \frac{4}{7}$ . Same degree: ratio of leading coefficients $\frac{4}{7}$ .

Flashcard 14: Determine the horizontal asymptote of $f(x) = \frac{x+2}{x^2+3x+1}$ .

Answer: Horizontal asymptote at $y = 0$ . Numerator degree less than denominator degree.

Flashcard 15: What is the end behavior of $f(x) = \frac{3x^4}{x^2 + 1}$ as $x \to \infty$ ?

Answer: $f(x)$ approaches $\infty$ as $x \to \infty$ . Numerator degree exceeds denominator degree.

Flashcard 16: Determine the end behavior of $f(x) = \frac{x^2}{x^3+1}$ as $x \to -\infty$ .

Answer: $f(x)$ approaches $0$ as $x \to -\infty$ . Degree of numerator less than denominator.

Flashcard 17: What is the vertical asymptote of $f(x) = \frac{2x^2}{x^2 - 9}$ ?

Answer: Vertical asymptotes at $x = 3$ and $x = -3$ . Factor denominator: $(x-3)(x+3) = 0$ .

Flashcard 18: State the vertical asymptotes for $f(x) = \frac{x^2-4}{x^2-5x+6}$ .

Answer: Vertical asymptotes at $x = 3$ and $x = 2$ . Factor denominator: $(x-3)(x-2) = 0$ .

Flashcard 19: Identify the vertical asymptote for $f(x) = \frac{x+1}{x-3}$ .

Answer: Vertical asymptote at $x = 3$ . Set denominator equal to zero: $x - 3 = 0$ .

Flashcard 20: What does the end behavior of $f(x) = \frac{x^3}{x+1}$ as $x \to \infty$ approach?

Answer: $f(x)$ approaches $\infty$ as $x \to \infty$ . Numerator degree exceeds denominator degree.

Flashcard 21: State the vertical asymptotes for $f(x) = \frac{x^2-4}{x^2-5x+6}$ .

Answer: Vertical asymptotes at $x = 3$ and $x = 2$ . Factor denominator: $(x-3)(x-2) = 0$ .

Flashcard 22: What is the horizontal asymptote for $f(x) = \frac{5x+1}{2x-3}$ ?

Answer: Horizontal asymptote at $y = \frac{5}{2}$ . Same degree: ratio of leading coefficients $\frac{5}{2}$ .

Flashcard 23: What does the end behavior of $f(x) = \frac{x^3}{x+1}$ as $x \to \infty$ approach?

Answer: $f(x)$ approaches $\infty$ as $x \to \infty$ . Numerator degree exceeds denominator degree.

Flashcard 24: Find the vertical asymptote of $f(x) = \frac{x^2 + 1}{x - 5}$ .

Answer: Vertical asymptote at $x = 5$ . Set denominator equal to zero: $x - 5 = 0$ .

Flashcard 25: What is the end behavior of $f(x) = \frac{1}{x}$ as $x$ approaches $\infty$ ?

Answer: $f(x)$ approaches $0$ as $x \to \infty$ . As denominator grows, fraction approaches zero.

Flashcard 26: State the vertical asymptotes for $f(x) = \frac{1}{x^2 - 4x + 4}$ .

Answer: Vertical asymptote at $x = 2$ . Perfect square: $(x-2)^2 = 0$ gives $x = 2$ .

Flashcard 27: Identify the horizontal asymptote for $f(x) = \frac{5x^3}{3x^3 + 1}$ .

Answer: Horizontal asymptote at $y = \frac{5}{3}$ . Same degree: ratio of leading coefficients $\frac{5}{3}$ .

Flashcard 28: Identify the removable discontinuity of $f(x) = \frac{x^2-1}{x^2+x}$ .

Answer: Removable discontinuity at $x = -1$ . Factor: $\frac{(x-1)(x+1)}{x(x+1)}$ cancels $(x+1)$ .

Flashcard 29: What is the end behavior of $f(x) = \frac{2x^3}{x^2+1}$ as $x \to -\infty$ ?

Answer: $f(x)$ approaches $-\infty$ as $x \to -\infty$ . Numerator degree exceeds denominator, goes to $-\infty$ .

Flashcard 30: What is the end behavior of $f(x) = \frac{x^2 + 2x + 1}{x^2}$ as $x \to \infty$ ?

Answer: $f(x)$ approaches $1$ as $x \to \infty$ . Expand and simplify: $\frac{x^2+2x+1}{x^2} = 1 + \frac{2}{x} + \frac{1}{x^2}$ .

Flashcard 31: State the vertical asymptotes for $f(x) = \frac{x^2-1}{x^2-4}$ .

Answer: Vertical asymptotes at $x = 2$ and $x = -2$ . Factor denominator: $(x-2)(x+2) = 0$ .

Flashcard 32: Identify the removable discontinuity of $f(x) = \frac{x^2-1}{x^2+x}$ .

Answer: Removable discontinuity at $x = -1$ . Factor: $\frac{(x-1)(x+1)}{x(x+1)}$ cancels $(x+1)$ .

Flashcard 33: Identify the removable discontinuity of $f(x) = \frac{(x-2)(x+3)}{x-2}$ .

Answer: Removable discontinuity at $x = 2$ . Common factor $(x-2)$ cancels out.

Flashcard 34: What is the horizontal asymptote of $f(x) = \frac{x^2+1}{2x^2-3x+5}$ ?

Answer: Horizontal asymptote at $y = \frac{1}{2}$ . Same degree: ratio of leading coefficients $\frac{1}{2}$ .

Flashcard 35: What is the vertical asymptote of $f(x) = \frac{2x^2}{x^2 - 9}$ ?

Answer: Vertical asymptotes at $x = 3$ and $x = -3$ . Factor denominator: $(x-3)(x+3) = 0$ .

Flashcard 36: What is a removable discontinuity in rational functions?

Answer: A hole where $p(x)$ and $q(x)$ share a common factor. Common factors cancel, creating a hole.

Flashcard 37: Identify the vertical asymptote for $f(x) = \frac{x+1}{x-3}$ .

Answer: Vertical asymptote at $x = 3$ . Set denominator equal to zero: $x - 3 = 0$ .

Flashcard 38: What is the end behavior of $f(x) = \frac{2x^2}{x^3 + 1}$ as $x \to \infty$ ?

Answer: $f(x)$ approaches $0$ as $x \to \infty$ . Numerator degree less than denominator degree.

Flashcard 39: Determine the vertical asymptote of $f(x) = \frac{x}{x^2-2x-3}$ .

Answer: Vertical asymptotes at $x = 3$ and $x = -1$ . Factor denominator: $(x-3)(x+1) = 0$ .

Flashcard 40: What is a rational function?

Answer: 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.

Flashcard 41: Determine the horizontal asymptote of $f(x) = \frac{x+2}{x^2+3x+1}$ .

Answer: Horizontal asymptote at $y = 0$ . Numerator degree less than denominator degree.

Flashcard 42: What is the horizontal asymptote for $f(x) = \frac{2x}{x^2+3}$ ?

Answer: Horizontal asymptote at $y = 0$ . Numerator degree less than denominator degree.

Flashcard 43: State the horizontal asymptote for $f(x) = \frac{4x+3}{7x+8}$ .

Answer: Horizontal asymptote at $y = \frac{4}{7}$ . Same degree: ratio of leading coefficients $\frac{4}{7}$ .

Flashcard 44: Determine the vertical asymptote of $f(x) = \frac{x}{x^2-2x-3}$ .

Answer: Vertical asymptotes at $x = 3$ and $x = -1$ . Factor denominator: $(x-3)(x+1) = 0$ .

Flashcard 45: What is the domain of $f(x) = \frac{1}{x^2 - 9}$ ?

Answer: All real numbers except $x = 3$ and $x = -3$ . Exclude values where $x^2 - 9 = 0$ .

Flashcard 46: What is the domain of $f(x) = \frac{5x}{x^2 - 4}$ ?

Answer: All real numbers except $x = 2$ and $x = -2$ . Exclude values where $x^2 - 4 = 0$ .

Flashcard 47: Identify the horizontal asymptote for $f(x) = \frac{2x^2}{x^2 + 1}$ .

Answer: Horizontal asymptote at $y = 2$ . Same degree: ratio of leading coefficients $\frac{2}{1}$ .

Flashcard 48: What is the domain of $f(x) = \frac{2}{x-4}$ ?

Answer: All real numbers except $x = 4$ . Exclude values where denominator equals zero.

Flashcard 49: What is the slant asymptote of $f(x) = \frac{x^2 + 2x + 3}{x - 1}$ ?

Answer: Slant asymptote is $y = x + 3$ . Numerator degree exceeds denominator by one.

Flashcard 50: What is the domain of $f(x) = \frac{5x}{x^2 - 4}$ ?

Answer: All real numbers except $x = 2$ and $x = -2$ . Exclude values where $x^2 - 4 = 0$ .

Flashcard 51: What is a rational function?

Answer: 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.

Flashcard 52: What is the domain of $f(x) = \frac{1}{x^2 - 9}$ ?

Answer: All real numbers except $x = 3$ and $x = -3$ . Exclude values where $x^2 - 9 = 0$ .

Flashcard 53: What is the end behavior of $f(x) = \frac{1}{x}$ as $x$ approaches $\infty$ ?

Answer: $f(x)$ approaches $0$ as $x \to \infty$ . As denominator grows, fraction approaches zero.

Flashcard 54: Identify the horizontal asymptote for $f(x) = \frac{2x^2}{x^2 + 1}$ .

Answer: Horizontal asymptote at $y = 2$ . Same degree: ratio of leading coefficients $\frac{2}{1}$ .

Flashcard 55: Identify the removable discontinuity of $f(x) = \frac{(x-2)(x+3)}{x-2}$ .

Answer: Removable discontinuity at $x = 2$ . Common factor $(x-2)$ cancels out.

Flashcard 56: Does $f(x) = \frac{x^2-4}{x^2+4}$ have a horizontal asymptote?

Answer: Yes, $y = 1$ . Same degree polynomials have horizontal asymptote.

Flashcard 57: What is the horizontal asymptote for $f(x) = \frac{5x+1}{2x-3}$ ?

Answer: Horizontal asymptote at $y = \frac{5}{2}$ . Same degree: ratio of leading coefficients $\frac{5}{2}$ .

Flashcard 58: Does $f(x) = \frac{x^2-4}{x^2+4}$ have a horizontal asymptote?

Answer: Yes, $y = 1$ . Same degree polynomials have horizontal asymptote.

Flashcard 59: State the horizontal asymptote of $f(x) = \frac{2x^2+3}{x^2-1}$ .

Answer: Horizontal asymptote at $y = 2$ . Same degree polynomials: ratio of leading coefficients.

Flashcard 60: What is the domain of $f(x) = \frac{x^2 + 3x - 4}{x^2 - 4}$ ?

Answer: All real numbers except $x = 2$ and $x = -2$ . Exclude values where $x^2 - 4 = 0$ .

Flashcard 61: What is the slant asymptote of $f(x) = \frac{x^2 + 2x + 3}{x - 1}$ ?

Answer: Slant asymptote is $y = x + 3$ . Numerator degree exceeds denominator by one.

Flashcard 62: What is the domain of $f(x) = \frac{x^2 + 3x - 4}{x^2 - 4}$ ?

Answer: All real numbers except $x = 2$ and $x = -2$ . Exclude values where $x^2 - 4 = 0$ .

Flashcard 63: Identify the horizontal asymptote for $f(x) = \frac{5x^3}{3x^3 + 1}$ .

Answer: Horizontal asymptote at $y = \frac{5}{3}$ . Same degree: ratio of leading coefficients $\frac{5}{3}$ .

Flashcard 64: Determine the vertical asymptote of $f(x) = \frac{x^2 + 1}{x^2 - 4x + 3}$ .

Answer: Vertical asymptotes at $x = 3$ and $x = 1$ . Factor denominator: $(x-3)(x-1) = 0$ .

Flashcard 65: What is the end behavior of $f(x) = \frac{3x^4}{x^2 + 1}$ as $x \to \infty$ ?

Answer: $f(x)$ approaches $\infty$ as $x \to \infty$ . Numerator degree exceeds denominator degree.

Flashcard 66: Determine the end behavior of $f(x) = \frac{x^2}{x^3+1}$ as $x \to -\infty$ .

Answer: $f(x)$ approaches $0$ as $x \to -\infty$ . Degree of numerator less than denominator.

Flashcard 67: What is the end behavior of $f(x) = \frac{3x^3}{2x^3 + 5}$ as $x$ approaches infinity?

Answer: $f(x)$ approaches $\frac{3}{2}$ as $x \to \infty$ . Same degree: divide leading coefficients $\frac{3}{2}$ .

Flashcard 68: What is the horizontal asymptote of $f(x) = \frac{x^2+1}{2x^2-3x+5}$ ?

Answer: Horizontal asymptote at $y = \frac{1}{2}$ . Same degree: ratio of leading coefficients $\frac{1}{2}$ .

Flashcard 69: What is the horizontal asymptote of $f(x) = \frac{3x}{x + 4}$ ?

Answer: Horizontal asymptote at $y = 3$ . Same degree: ratio of leading coefficients $\frac{3}{1}$ .

AP Precalculus Flashcards: Rational Functions And End Behavior

Rational Functions And End Behavior

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: State the vertical asymptotes for f(x)=1x2−4x+4f(x) = \frac{1}{x^2 - 4x + 4}f(x)=x2−4x+41​.

Flashcard 2: What is the end behavior of f(x)=3x32x3+5f(x) = \frac{3x^3}{2x^3 + 5}f(x)=2x3+53x3​ as xxx approaches infinity?

Flashcard 3: State the vertical asymptotes for f(x)=x2−1x2−4f(x) = \frac{x^2-1}{x^2-4}f(x)=x2−4x2−1​.

Flashcard 4: What is a removable discontinuity in rational functions?

Flashcard 5: What is the end behavior of f(x)=2x2x3+1f(x) = \frac{2x^2}{x^3 + 1}f(x)=x3+12x2​ as x→∞x \to \inftyx→∞?

Flashcard 6: State the horizontal asymptote of f(x)=2x2+3x2−1f(x) = \frac{2x^2+3}{x^2-1}f(x)=x2−12x2+3​.

Flashcard 7: What is the horizontal asymptote of f(x)=3xx+4f(x) = \frac{3x}{x + 4}f(x)=x+43x​?

Flashcard 8: Determine the vertical asymptote of f(x)=x2+1x2−4x+3f(x) = \frac{x^2 + 1}{x^2 - 4x + 3}f(x)=x2−4x+3x2+1​.

Flashcard 9: What is the horizontal asymptote for f(x)=2xx2+3f(x) = \frac{2x}{x^2+3}f(x)=x2+32x​?

Flashcard 10: What is the end behavior of f(x)=x2+2x+1x2f(x) = \frac{x^2 + 2x + 1}{x^2}f(x)=x2x2+2x+1​ as x→∞x \to \inftyx→∞?

Flashcard 11: What is the end behavior of f(x)=2x3x2+1f(x) = \frac{2x^3}{x^2+1}f(x)=x2+12x3​ as x→−∞x \to -\inftyx→−∞?

Flashcard 12: Find the vertical asymptote of f(x)=x2+1x−5f(x) = \frac{x^2 + 1}{x - 5}f(x)=x−5x2+1​.

Flashcard 13: State the horizontal asymptote for f(x)=4x+37x+8f(x) = \frac{4x+3}{7x+8}f(x)=7x+84x+3​.

Flashcard 14: Determine the horizontal asymptote of f(x)=x+2x2+3x+1f(x) = \frac{x+2}{x^2+3x+1}f(x)=x2+3x+1x+2​.

Flashcard 15: What is the end behavior of f(x)=3x4x2+1f(x) = \frac{3x^4}{x^2 + 1}f(x)=x2+13x4​ as x→∞x \to \inftyx→∞?

Flashcard 16: Determine the end behavior of f(x)=x2x3+1f(x) = \frac{x^2}{x^3+1}f(x)=x3+1x2​ as x→−∞x \to -\inftyx→−∞.

Flashcard 17: What is the vertical asymptote of f(x)=2x2x2−9f(x) = \frac{2x^2}{x^2 - 9}f(x)=x2−92x2​?

Flashcard 18: State the vertical asymptotes for f(x)=x2−4x2−5x+6f(x) = \frac{x^2-4}{x^2-5x+6}f(x)=x2−5x+6x2−4​.

Flashcard 19: Identify the vertical asymptote for f(x)=x+1x−3f(x) = \frac{x+1}{x-3}f(x)=x−3x+1​.

Flashcard 20: What does the end behavior of f(x)=x3x+1f(x) = \frac{x^3}{x+1}f(x)=x+1x3​ as x→∞x \to \inftyx→∞ approach?

Flashcard 21: State the vertical asymptotes for f(x)=x2−4x2−5x+6f(x) = \frac{x^2-4}{x^2-5x+6}f(x)=x2−5x+6x2−4​.

Flashcard 22: What is the horizontal asymptote for f(x)=5x+12x−3f(x) = \frac{5x+1}{2x-3}f(x)=2x−35x+1​?

Flashcard 23: What does the end behavior of f(x)=x3x+1f(x) = \frac{x^3}{x+1}f(x)=x+1x3​ as x→∞x \to \inftyx→∞ approach?

Flashcard 24: Find the vertical asymptote of f(x)=x2+1x−5f(x) = \frac{x^2 + 1}{x - 5}f(x)=x−5x2+1​.

Flashcard 25: What is the end behavior of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ as xxx approaches ∞\infty∞?

Flashcard 26: State the vertical asymptotes for f(x)=1x2−4x+4f(x) = \frac{1}{x^2 - 4x + 4}f(x)=x2−4x+41​.

Flashcard 27: Identify the horizontal asymptote for f(x)=5x33x3+1f(x) = \frac{5x^3}{3x^3 + 1}f(x)=3x3+15x3​.

Flashcard 28: Identify the removable discontinuity of f(x)=x2−1x2+xf(x) = \frac{x^2-1}{x^2+x}f(x)=x2+xx2−1​.

Flashcard 29: What is the end behavior of f(x)=2x3x2+1f(x) = \frac{2x^3}{x^2+1}f(x)=x2+12x3​ as x→−∞x \to -\inftyx→−∞?

Flashcard 30: What is the end behavior of f(x)=x2+2x+1x2f(x) = \frac{x^2 + 2x + 1}{x^2}f(x)=x2x2+2x+1​ as x→∞x \to \inftyx→∞?

Flashcard 31: State the vertical asymptotes for f(x)=x2−1x2−4f(x) = \frac{x^2-1}{x^2-4}f(x)=x2−4x2−1​.

Flashcard 32: Identify the removable discontinuity of f(x)=x2−1x2+xf(x) = \frac{x^2-1}{x^2+x}f(x)=x2+xx2−1​.

Flashcard 33: Identify the removable discontinuity of f(x)=(x−2)(x+3)x−2f(x) = \frac{(x-2)(x+3)}{x-2}f(x)=x−2(x−2)(x+3)​.

Flashcard 34: What is the horizontal asymptote of f(x)=x2+12x2−3x+5f(x) = \frac{x^2+1}{2x^2-3x+5}f(x)=2x2−3x+5x2+1​?

Flashcard 35: What is the vertical asymptote of f(x)=2x2x2−9f(x) = \frac{2x^2}{x^2 - 9}f(x)=x2−92x2​?

Flashcard 36: What is a removable discontinuity in rational functions?

Flashcard 37: Identify the vertical asymptote for f(x)=x+1x−3f(x) = \frac{x+1}{x-3}f(x)=x−3x+1​.

Flashcard 38: What is the end behavior of f(x)=2x2x3+1f(x) = \frac{2x^2}{x^3 + 1}f(x)=x3+12x2​ as x→∞x \to \inftyx→∞?

Flashcard 39: Determine the vertical asymptote of f(x)=xx2−2x−3f(x) = \frac{x}{x^2-2x-3}f(x)=x2−2x−3x​.

Flashcard 40: What is a rational function?

Flashcard 41: Determine the horizontal asymptote of f(x)=x+2x2+3x+1f(x) = \frac{x+2}{x^2+3x+1}f(x)=x2+3x+1x+2​.

Flashcard 42: What is the horizontal asymptote for f(x)=2xx2+3f(x) = \frac{2x}{x^2+3}f(x)=x2+32x​?

Flashcard 43: State the horizontal asymptote for f(x)=4x+37x+8f(x) = \frac{4x+3}{7x+8}f(x)=7x+84x+3​.

Flashcard 44: Determine the vertical asymptote of f(x)=xx2−2x−3f(x) = \frac{x}{x^2-2x-3}f(x)=x2−2x−3x​.

Flashcard 45: What is the domain of f(x)=1x2−9f(x) = \frac{1}{x^2 - 9}f(x)=x2−91​?

Flashcard 46: What is the domain of f(x)=5xx2−4f(x) = \frac{5x}{x^2 - 4}f(x)=x2−45x​?

Flashcard 47: Identify the horizontal asymptote for f(x)=2x2x2+1f(x) = \frac{2x^2}{x^2 + 1}f(x)=x2+12x2​.

Flashcard 48: What is the domain of f(x)=2x−4f(x) = \frac{2}{x-4}f(x)=x−42​?

Flashcard 49: What is the slant asymptote of f(x)=x2+2x+3x−1f(x) = \frac{x^2 + 2x + 3}{x - 1}f(x)=x−1x2+2x+3​?

Flashcard 50: What is the domain of f(x)=5xx2−4f(x) = \frac{5x}{x^2 - 4}f(x)=x2−45x​?

Flashcard 51: What is a rational function?

Flashcard 52: What is the domain of f(x)=1x2−9f(x) = \frac{1}{x^2 - 9}f(x)=x2−91​?

Flashcard 53: What is the end behavior of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ as xxx approaches ∞\infty∞?

Flashcard 54: Identify the horizontal asymptote for f(x)=2x2x2+1f(x) = \frac{2x^2}{x^2 + 1}f(x)=x2+12x2​.

Flashcard 55: Identify the removable discontinuity of f(x)=(x−2)(x+3)x−2f(x) = \frac{(x-2)(x+3)}{x-2}f(x)=x−2(x−2)(x+3)​.

Flashcard 56: Does f(x)=x2−4x2+4f(x) = \frac{x^2-4}{x^2+4}f(x)=x2+4x2−4​ have a horizontal asymptote?

Flashcard 57: What is the horizontal asymptote for f(x)=5x+12x−3f(x) = \frac{5x+1}{2x-3}f(x)=2x−35x+1​?

Flashcard 58: Does f(x)=x2−4x2+4f(x) = \frac{x^2-4}{x^2+4}f(x)=x2+4x2−4​ have a horizontal asymptote?

Flashcard 59: State the horizontal asymptote of f(x)=2x2+3x2−1f(x) = \frac{2x^2+3}{x^2-1}f(x)=x2−12x2+3​.

Flashcard 60: What is the domain of f(x)=x2+3x−4x2−4f(x) = \frac{x^2 + 3x - 4}{x^2 - 4}f(x)=x2−4x2+3x−4​?

Flashcard 61: What is the slant asymptote of f(x)=x2+2x+3x−1f(x) = \frac{x^2 + 2x + 3}{x - 1}f(x)=x−1x2+2x+3​?

Flashcard 62: What is the domain of f(x)=x2+3x−4x2−4f(x) = \frac{x^2 + 3x - 4}{x^2 - 4}f(x)=x2−4x2+3x−4​?

Flashcard 63: Identify the horizontal asymptote for f(x)=5x33x3+1f(x) = \frac{5x^3}{3x^3 + 1}f(x)=3x3+15x3​.

Flashcard 64: Determine the vertical asymptote of f(x)=x2+1x2−4x+3f(x) = \frac{x^2 + 1}{x^2 - 4x + 3}f(x)=x2−4x+3x2+1​.

Flashcard 65: What is the end behavior of f(x)=3x4x2+1f(x) = \frac{3x^4}{x^2 + 1}f(x)=x2+13x4​ as x→∞x \to \inftyx→∞?

Flashcard 66: Determine the end behavior of f(x)=x2x3+1f(x) = \frac{x^2}{x^3+1}f(x)=x3+1x2​ as x→−∞x \to -\inftyx→−∞.

Flashcard 67: What is the end behavior of f(x)=3x32x3+5f(x) = \frac{3x^3}{2x^3 + 5}f(x)=2x3+53x3​ as xxx approaches infinity?

Flashcard 68: What is the horizontal asymptote of f(x)=x2+12x2−3x+5f(x) = \frac{x^2+1}{2x^2-3x+5}f(x)=2x2−3x+5x2+1​?

Flashcard 69: What is the horizontal asymptote of f(x)=3xx+4f(x) = \frac{3x}{x + 4}f(x)=x+43x​?

Flashcard 1: State the vertical asymptotes for $f(x) = \frac{1}{x^2 - 4x + 4}$ .

Flashcard 2: What is the end behavior of $f(x) = \frac{3x^3}{2x^3 + 5}$ as $x$ approaches infinity?

Flashcard 3: State the vertical asymptotes for $f(x) = \frac{x^2-1}{x^2-4}$ .

Flashcard 5: What is the end behavior of $f(x) = \frac{2x^2}{x^3 + 1}$ as $x \to \infty$ ?

Flashcard 6: State the horizontal asymptote of $f(x) = \frac{2x^2+3}{x^2-1}$ .

Flashcard 7: What is the horizontal asymptote of $f(x) = \frac{3x}{x + 4}$ ?

Flashcard 8: Determine the vertical asymptote of $f(x) = \frac{x^2 + 1}{x^2 - 4x + 3}$ .

Flashcard 9: What is the horizontal asymptote for $f(x) = \frac{2x}{x^2+3}$ ?

Flashcard 10: What is the end behavior of $f(x) = \frac{x^2 + 2x + 1}{x^2}$ as $x \to \infty$ ?

Flashcard 11: What is the end behavior of $f(x) = \frac{2x^3}{x^2+1}$ as $x \to -\infty$ ?

Flashcard 12: Find the vertical asymptote of $f(x) = \frac{x^2 + 1}{x - 5}$ .

Flashcard 13: State the horizontal asymptote for $f(x) = \frac{4x+3}{7x+8}$ .

Flashcard 14: Determine the horizontal asymptote of $f(x) = \frac{x+2}{x^2+3x+1}$ .

Flashcard 15: What is the end behavior of $f(x) = \frac{3x^4}{x^2 + 1}$ as $x \to \infty$ ?

Flashcard 16: Determine the end behavior of $f(x) = \frac{x^2}{x^3+1}$ as $x \to -\infty$ .

Flashcard 17: What is the vertical asymptote of $f(x) = \frac{2x^2}{x^2 - 9}$ ?

Flashcard 18: State the vertical asymptotes for $f(x) = \frac{x^2-4}{x^2-5x+6}$ .

Flashcard 19: Identify the vertical asymptote for $f(x) = \frac{x+1}{x-3}$ .

Flashcard 20: What does the end behavior of $f(x) = \frac{x^3}{x+1}$ as $x \to \infty$ approach?

Flashcard 21: State the vertical asymptotes for $f(x) = \frac{x^2-4}{x^2-5x+6}$ .

Flashcard 22: What is the horizontal asymptote for $f(x) = \frac{5x+1}{2x-3}$ ?

Flashcard 23: What does the end behavior of $f(x) = \frac{x^3}{x+1}$ as $x \to \infty$ approach?

Flashcard 24: Find the vertical asymptote of $f(x) = \frac{x^2 + 1}{x - 5}$ .

Flashcard 25: What is the end behavior of $f(x) = \frac{1}{x}$ as $x$ approaches $\infty$ ?

Flashcard 26: State the vertical asymptotes for $f(x) = \frac{1}{x^2 - 4x + 4}$ .

Flashcard 27: Identify the horizontal asymptote for $f(x) = \frac{5x^3}{3x^3 + 1}$ .

Flashcard 28: Identify the removable discontinuity of $f(x) = \frac{x^2-1}{x^2+x}$ .

Flashcard 29: What is the end behavior of $f(x) = \frac{2x^3}{x^2+1}$ as $x \to -\infty$ ?

Flashcard 30: What is the end behavior of $f(x) = \frac{x^2 + 2x + 1}{x^2}$ as $x \to \infty$ ?

Flashcard 31: State the vertical asymptotes for $f(x) = \frac{x^2-1}{x^2-4}$ .

Flashcard 32: Identify the removable discontinuity of $f(x) = \frac{x^2-1}{x^2+x}$ .

Flashcard 33: Identify the removable discontinuity of $f(x) = \frac{(x-2)(x+3)}{x-2}$ .

Flashcard 34: What is the horizontal asymptote of $f(x) = \frac{x^2+1}{2x^2-3x+5}$ ?

Flashcard 35: What is the vertical asymptote of $f(x) = \frac{2x^2}{x^2 - 9}$ ?

Flashcard 37: Identify the vertical asymptote for $f(x) = \frac{x+1}{x-3}$ .

Flashcard 38: What is the end behavior of $f(x) = \frac{2x^2}{x^3 + 1}$ as $x \to \infty$ ?

Flashcard 39: Determine the vertical asymptote of $f(x) = \frac{x}{x^2-2x-3}$ .

Flashcard 41: Determine the horizontal asymptote of $f(x) = \frac{x+2}{x^2+3x+1}$ .

Flashcard 42: What is the horizontal asymptote for $f(x) = \frac{2x}{x^2+3}$ ?

Flashcard 43: State the horizontal asymptote for $f(x) = \frac{4x+3}{7x+8}$ .

Flashcard 44: Determine the vertical asymptote of $f(x) = \frac{x}{x^2-2x-3}$ .

Flashcard 45: What is the domain of $f(x) = \frac{1}{x^2 - 9}$ ?

Flashcard 46: What is the domain of $f(x) = \frac{5x}{x^2 - 4}$ ?

Flashcard 47: Identify the horizontal asymptote for $f(x) = \frac{2x^2}{x^2 + 1}$ .

Flashcard 48: What is the domain of $f(x) = \frac{2}{x-4}$ ?

Flashcard 49: What is the slant asymptote of $f(x) = \frac{x^2 + 2x + 3}{x - 1}$ ?

Flashcard 50: What is the domain of $f(x) = \frac{5x}{x^2 - 4}$ ?

Flashcard 52: What is the domain of $f(x) = \frac{1}{x^2 - 9}$ ?

Flashcard 53: What is the end behavior of $f(x) = \frac{1}{x}$ as $x$ approaches $\infty$ ?

Flashcard 54: Identify the horizontal asymptote for $f(x) = \frac{2x^2}{x^2 + 1}$ .

Flashcard 55: Identify the removable discontinuity of $f(x) = \frac{(x-2)(x+3)}{x-2}$ .

Flashcard 56: Does $f(x) = \frac{x^2-4}{x^2+4}$ have a horizontal asymptote?

Flashcard 57: What is the horizontal asymptote for $f(x) = \frac{5x+1}{2x-3}$ ?

Flashcard 58: Does $f(x) = \frac{x^2-4}{x^2+4}$ have a horizontal asymptote?

Flashcard 59: State the horizontal asymptote of $f(x) = \frac{2x^2+3}{x^2-1}$ .

Flashcard 60: What is the domain of $f(x) = \frac{x^2 + 3x - 4}{x^2 - 4}$ ?

Flashcard 61: What is the slant asymptote of $f(x) = \frac{x^2 + 2x + 3}{x - 1}$ ?

Flashcard 62: What is the domain of $f(x) = \frac{x^2 + 3x - 4}{x^2 - 4}$ ?

Flashcard 63: Identify the horizontal asymptote for $f(x) = \frac{5x^3}{3x^3 + 1}$ .

Flashcard 64: Determine the vertical asymptote of $f(x) = \frac{x^2 + 1}{x^2 - 4x + 3}$ .

Flashcard 65: What is the end behavior of $f(x) = \frac{3x^4}{x^2 + 1}$ as $x \to \infty$ ?

Flashcard 66: Determine the end behavior of $f(x) = \frac{x^2}{x^3+1}$ as $x \to -\infty$ .

Flashcard 67: What is the end behavior of $f(x) = \frac{3x^3}{2x^3 + 5}$ as $x$ approaches infinity?

Flashcard 68: What is the horizontal asymptote of $f(x) = \frac{x^2+1}{2x^2-3x+5}$ ?

Flashcard 69: What is the horizontal asymptote of $f(x) = \frac{3x}{x + 4}$ ?