Rational Functions and Zeros - AP Precalculus

1. Identify a removable discontinuity in $f(x) = \frac{x^2 - 1}{x - 1}$.

   Removable discontinuity at $x = 1$. The factor $(x-1)$ cancels from numerator and denominator.

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

   The slant asymptote is $y = x$. Divide $x^2 + 1$ by $x$ using polynomial long division.

3. What is the horizontal asymptote of $f(x) = \frac{7x}{2x + 3}$?

   The horizontal asymptote is $y = \frac{7}{2}$. Equal degrees give horizontal asymptote $y = \frac{7}{2}$.

4. Identify the hole in $f(x) = \frac{x^2 - 1}{x - 1}$.

   Hole at $x = 1$. Factor $x^2 - 1 = (x-1)(x+1)$ and cancel the common factor.

5. What is a zero of a rational function?

   A value of $x$ for which $f(x) = 0$. A zero occurs when the numerator equals zero but the denominator doesn't.

6. What is the degree of the polynomial $p(x) = 3x^4 - 2x^2 + 1$?

   The degree is 4. The highest power term determines the degree of a polynomial.

7. Find the vertical asymptote of $f(x) = \frac{2x + 3}{x - 4}$.

   Vertical asymptote at $x = 4$. Set the denominator $x - 4 = 0$ to find the vertical asymptote.

8. Identify the vertical asymptote of $f(x) = \frac{2x}{x^2 - 4}$.

   Vertical asymptotes are $x = 2$ and $x = -2$. Factor $x^2 - 4 = (x-2)(x+2)$ to find where the denominator is zero.

9. What is the behavior of $f(x) = \frac{1}{x}$ as $x \to 0$?

   As $x \to 0$, $f(x) \to \pm \infty$. The function has a vertical asymptote at $x = 0$.

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

    The horizontal asymptote is $y = 1$. When degrees are equal, the horizontal asymptote is $\frac{1}{1} = 1$.

11. What is the horizontal asymptote for $f(x) = \frac{2x^2}{x^2 + 1}$?

    The horizontal asymptote is $y = 2$. Equal degrees give horizontal asymptote $y = \frac{2}{1} = 2$.

12. Which term denotes values not in the domain of a rational function?

    These are the poles or vertical asymptotes. Points where the denominator equals zero make the function undefined.

13. State the horizontal asymptote for $f(x) = \frac{4x}{x + 2}$.

    The horizontal asymptote is $y = 4$. As $x \to \infty$, the function approaches $\frac{4}{1} = 4$.

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

    Domain: $x \neq 0, x \neq 4$. Factor the denominator: $x^2 - 4x = x(x - 4)$.

15. What is the behavior of $f(x) = \frac{x}{x^2 - 1}$ as $x \to 1$?

    As $x \to 1$, $f(x) \to \infty$. The denominator $(x-1)$ approaches 0 while numerator approaches 1.

16. What are the zeros of $f(x) = \frac{x^2 - 9}{x + 2}$?

    The zeros are $x = 3$ and $x = -3$. Factor the numerator: $x^2 - 9 = (x-3)(x+3)$.

17. State the condition for a vertical asymptote in a rational function.

    Occurs where $q(x) = 0$ and $p(x) \neq 0$. The denominator is zero while the numerator is non-zero.

18. Which term describes $x$ values making $f(x)$ undefined?

    These are the excluded values or domain restrictions. Values that make the denominator zero are excluded from the domain.

19. Find the zero of $f(x) = \frac{2x - 4}{x + 3}$.

    The zero is $x = 2$. Set $2x - 4 = 0$ to find where the numerator equals zero.

20. Find the domain of $f(x) = \frac{4x + 1}{x^2 - 9}$.

    Domain: $x \neq 3, x \neq -3$. Set $x^2 - 9 = 0$ to find where the function is undefined.

21. Identify the horizontal asymptote of $f(x) = \frac{5x}{x^2 + 4}$.

    The horizontal asymptote is $y = 0$. Numerator degree is less than denominator degree, so $y = 0$.

22. What is the degree of $q(x) = x^3 - 7x$?

    The degree is 3. The highest power term $x^3$ has degree 3.

23. Find the zeros of $f(x) = \frac{3x^2 - 12}{x + 2}$.

    The zeros are $x = 2$ and $x = -2$. Factor the numerator: $3x^2 - 12 = 3(x^2 - 4) = 3(x-2)(x+2)$.

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

    The horizontal asymptote is $y = 2$. Equal degrees give horizontal asymptote $y = \frac{2}{1} = 2$.

25. Find the zero of $f(x) = \frac{x - 4}{x + 1}$.

    The zero is $x = 4$. Set the numerator $x - 4 = 0$ to find where $f(x) = 0$.

26. State the vertical asymptote of $f(x) = \frac{2x + 1}{x^2 - 16}$.

    Vertical asymptotes at $x = 4$ and $x = -4$. Factor the denominator: $x^2 - 16 = (x-4)(x+4)$.

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

    The horizontal asymptote is $y = 1$. When degrees are equal, the horizontal asymptote equals the ratio of leading coefficients.

28. What is the end behavior of $f(x) = \frac{3x^3 + 4}{5x^3 + 2}$?

    As $x \to \pm \infty$, $f(x) \to \frac{3}{5}$. When degrees are equal, divide leading coefficients: $\frac{3}{5}$.

29. What is the behavior of $f(x) = \frac{2x}{x + 5}$ as $x \to -5$?

    As $x \to -5$, $f(x) \to \pm \infty$. The function has a vertical asymptote at $x = -5$.

30. Identify the vertical asymptote of $f(x) = \frac{5}{x^2 - 25}$.

    Vertical asymptotes at $x = 5$ and $x = -5$. Factor the denominator: $x^2 - 25 = (x-5)(x+5)$.

31. For $f(x) = \frac{x^2 - 4}{x - 2}$, what is the discontinuity?

    Discontinuity (hole) at $x = 2$. Factor and cancel: $(x-2)$ appears in both numerator and denominator.

32. What is the intercept of $f(x) = \frac{3}{x-2}$?

    Vertical intercept at $(0, -\frac{3}{2})$. Substitute $x = 0$ into the function to find the y-intercept.

33. How do you determine a hole in a rational function?

    If $p(x)$ and $q(x)$ share a common factor. Common factors can be cancelled, creating a removable discontinuity.

34. Describe when a rational function has no horizontal asymptote.

    When the degree of $p(x)$ is greater than $q(x)$. When numerator degree exceeds denominator degree, no horizontal asymptote exists.

35. What happens to $f(x) = \frac{1}{x-3}$ as $x \to 3$?

    $f(x) \to \pm \infty$, vertical asymptote at $x = 3$. As $x$ approaches 3, the denominator approaches 0 while numerator stays 1.

36. What is the general form of a rational function?

    $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials. This defines a rational function as a ratio of two polynomial functions.

37. What is the y-intercept of $f(x) = \frac{2x - 5}{x + 3}$?

    The y-intercept is $(0, -\frac{5}{3})$. Substitute $x = 0$ to get $f(0) = \frac{-5}{3}$.

38. What is the domain of $f(x) = \frac{1}{x(x + 3)}$?

    Domain: $x \neq 0, x \neq -3$. Set $x(x + 3) = 0$ to find where the function is undefined.

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