Rational Functions and Holes - AP Precalculus

1. Perform long division on $f(x) = \frac{x^2 + 3x + 2}{x + 1}$.

   Quotient is $x + 2$, remainder is 0. $x^2 + 3x + 2 = (x+1)(x+2)$ divides evenly.

2. Find the removable discontinuity of $f(x) = \frac{x^2 - 9}{x - 3}$.

   Removable discontinuity at $x = 3$. $\frac{(x-3)(x+3)}{x-3}$ cancels at $x = 3$.

3. What role do intercepts play in graphing rational functions?

   Determine where the graph crosses axes. Show where function crosses or touches coordinate axes.

4. State the behavior near a hole in a rational function.

   Function approaches a limit but is not defined at the hole. Limiting value exists despite the discontinuity.

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

   Horizontal asymptote at $y = 0$. Denominator degree exceeds numerator, so limit is zero.

6. What is the end behavior of a rational function?

   Describes $f(x)$ as $x \to \pm\infty$, often related to asymptotes. Behavior of function values as $x$ approaches infinity.

7. Identify the hole in $f(x) = \frac{x^2 - 25}{x^2 - 5x}$.

   Hole at $x = 5$. $\frac{(x-5)(x+5)}{x(x-5)}$ has $(x-5)$ common factor.

8. What happens to $f(x) = \frac{x^2 - 1}{x - 1}$ near $x = 1$?

   Approaches $x + 1$, but undefined at $x = 1$. $\frac{(x-1)(x+1)}{x-1}$ approaches $x+1 = 2$ as $x \to 1$.

9. What are the steps to identify holes in a rational function?

   Factor $p(x)$ and $q(x)$, find common factors. Cancel common factors to reveal removable discontinuities.

10. Simplify $f(x) = \frac{x^2 - 4x + 4}{x - 2}$.

    $f(x) = x - 2$ for $x \neq 2$. $\frac{(x-2)^2}{x-2}$ simplifies to $(x-2)$ with hole.

11. What is the effect of a higher-degree $p(x)$ on end behavior?

    Dominates $q(x)$, leading to polynomial-like behavior. Function grows without bound as $x$ increases.

12. Identify the condition for a hole in a rational function.

    A hole occurs if $p(x)$ and $q(x)$ have a common factor. Common factors cancel, creating removable discontinuities.

13. What is the $y$-intercept of a rational function $f(x) = \frac{p(x)}{q(x)}$?

    $f(0) = \frac{p(0)}{q(0)}$, if defined. Evaluate function at $x = 0$ if denominator nonzero.

14. What is the result of long division of $p(x)$ by $q(x)$ in rational functions?

    Quotient determines slant asymptote if degrees differ by 1. Provides slant asymptote when numerator degree exceeds by one.

15. What defines a vertical asymptote in a rational function?

    Occurs at $x$ where $q(x) = 0$ and $p(x)$ does not cancel. Denominator zero with no cancellation creates vertical line.

16. Find the hole of $f(x) = \frac{x^2 - 4}{x - 2}$.

    Hole at $x = 2$, as $x - 2$ is a common factor. Factor: $\frac{(x-2)(x+2)}{x-2}$, so $(x-2)$ cancels.

17. What does it mean for a rational function to be simplified?

    No common factors between $p(x)$ and $q(x)$. All common factors have been canceled from the fraction.

18. When does a rational function have no horizontal asymptote?

    When degree of $p(x)$ is greater than $q(x)$. Numerator degree exceeds denominator degree.

19. Define the term 'non-removable discontinuity.'

    A vertical asymptote in a rational function. Infinite discontinuity that cannot be removed by cancellation.

20. Describe the end behavior of $f(x) = \frac{x^3}{x^2}$.

    As $x \to \pm\infty$, $f(x) \to \pm\infty$. $\frac{x^3}{x^2} = x$ grows linearly to infinity.

21. Which values of $x$ cause holes in $f(x) = \frac{x^2 - 1}{x^2 - 2x + 1}$?

    Hole at $x = 1$, as $(x-1)$ is a common factor. $\frac{(x-1)(x+1)}{(x-1)^2}$ has $(x-1)$ in common.

22. What is the impact of a common factor on the graph of a rational function?

    Creates a hole in the graph at the factor's root. Removable discontinuity where factors cancel out.

23. State the removable discontinuity in a rational function.

    A point where the function is not defined due to a hole. Gap in graph that can be 'filled' by canceling factors.

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

    Vertical asymptotes at $x = 1$ and $x = -1$. $x^2 - 1 = (x-1)(x+1) = 0$ when $x = \pm 1$.

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

    Slant asymptote is $y = x + 1$. Divide $x^2 + 1$ by $x - 1$ using long division.

26. Identify the end behavior of $f(x) = \frac{x^3}{x^2 + 1}$.

    As $x \to \pm\infty$, $f(x) \to \pm\infty$. Numerator degree exceeds denominator, so no horizontal limit.

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

    Horizontal asymptote at $y = \frac{3}{2}$. Equal degrees: ratio of leading coefficients is $\frac{3}{2}$.

28. Find the $x$-intercept of $f(x) = \frac{x^2 - 9}{x + 3}$.

    $x$-intercept at $x = 3$. $\frac{(x-3)(x+3)}{x+3}$ has zero at $x = 3$.

29. Find the common factor of $f(x) = \frac{x^2 - 4x}{x(x-4)}$.

    Common factor is $x$. Factor $x$ appears in both numerator and denominator.

30. Identify the vertical asymptote of $f(x) = \frac{2x}{x^2 - 9}$.

    Vertical asymptotes at $x = 3$ and $x = -3$. $x^2 - 9 = (x-3)(x+3) = 0$ when $x = \pm 3$.

31. State the domain of a rational function.

    All real numbers except where $q(x) = 0$. Excludes values that make the denominator zero.

32. What is a horizontal asymptote for a rational function?

    Occurs as $x \to \infty$, based on the degrees of $p(x)$ and $q(x)$. Determined by comparing degrees of numerator and denominator.

33. What is the significance of the leading coefficients in asymptotes?

    They determine the horizontal asymptote when degrees are equal. When degrees are equal, their ratio gives horizontal asymptote.

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

    Horizontal asymptote at $y = 2$. Equal degrees: $\frac{4}{2} = 2$ gives horizontal asymptote.

35. Define a slant (oblique) asymptote in a rational function.

    Occurs when degree of $p(x)$ is one more than $q(x)$. Creates diagonal asymptote from polynomial long division.

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. Standard notation where both numerator and denominator are polynomials.

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