Rational Functions and End Behavior - AP Precalculus

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

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

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

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

3. What is a removable discontinuity in rational functions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

19. What is a rational function?

    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.

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

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

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

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

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

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

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

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

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

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

25. What is the domain of $f(x) = \frac{2}{x-4}$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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