5.7.1: Resources and Key Concepts

Resources

Videos

• Rational function
  • Rational Functions - A Brief Review of Rational Functions
• Vertical asymptote
  • Rational Functions - Identifying Vertical Asymptotes of Rational Functions
  • Rational Functions - Asymptotic Behavior of Rational Functions
• Horizontal asymptote
  • Rational Functions - Identifying Horizontal Asymptotes of Rational Functions
• Slant asymptote (oblique asymptote)
  • Rational Functions - Identifying Slant Asymptotes of Rational Functions
• Domain of a rational function
  • Rational Functions - Finding the Domain of Rational Functions
• Graphing rational functions
  • Rational Functions - Graphing Rational Functions
  • Rational Functions - Graphing Rational Functions
  • Algebra Lecture: Graphing Rational Functions
• Solving rational inequalities
  • Rational Functions - Solving Rational Inequalities
  • Inequalities - Nonlinear - A Review for College Algebra: Solving Rational Inequalities

Key Concepts

Definitions

• Arrow Notation: A symbolic way to describe the behavior of a function as its input \(x\) approaches a certain value or infinity. Examples: \(x \rightarrow a^-\) (approaches \(a\) from the left), \(x \rightarrow \infty\) (approaches infinity), \(f(x) \rightarrow L\) (the output approaches \(L\)).
• Vertical Asymptote: A vertical line \(x=a\) where the graph of a function tends toward positive or negative infinity as the inputs approach \(a\) from either the left or the right.
• Horizontal Asymptote: A horizontal line \(y=b\) where the graph of a function approaches the line as the inputs increase or decrease without bound (\(x \rightarrow \infty\) or \(x \rightarrow -\infty\)).
• Rational Function: A function that can be written as the quotient of two polynomial functions, \(f(x) = \frac{P(x)}{Q(x)}\), where \(Q(x) \neq 0\).
• Removable Discontinuity (Hole): A single point where the graph of a function is not defined, indicated by an open circle on the graph. It occurs in a rational function at \(x=a\) if \((x-a)\) is a factor of both the numerator and the denominator, and after simplification, \(a\) is no longer a zero of the denominator.
• End Behavior (of a rational function's graph): A description, usually as a function (e.g., horizontal or slant asymptote), of what the function values of the graph tend to approach as the inputs increase or decrease without bound.
• Polynomial Degree Notation (\(deg(f(x))\)): Denotes the degree of a polynomial \(f(x)\).
• Slant Asymptote (Oblique Asymptote): A non-horizontal, non-vertical line that the graph of a rational function approaches as \(x \rightarrow \infty\) or \(x \rightarrow -\infty\). It occurs when the degree of the numerator is exactly one more than the degree of the denominator.

Theorems

• Domain of a Rational Function: The domain of a rational function includes all real numbers except those that cause the denominator of the unsimplified rational function to equal zero.
• Vertical Asymptotes of Rational Functions: Vertical asymptotes of a rational function may be found by examining the factors of the denominator that are not common to the factors in the numerator. Vertical asymptotes occur at the zeros of such factors.
• Removable Discontinuity of a Rational Function: A removable discontinuity for the graph of a rational function occurs at \(x=a\) if \(a\) is a zero for a factor in the denominator that is common with a factor in the numerator, and after simplification, \(a\) is no longer a zero of the denominator.
• End Behavior of a Rational Function \(f(x) = \frac{N(x)}{D(x)}\):
  • If \(deg(D(x)) > deg(N(x))\), then \(f(x)\) has a horizontal asymptote at \(y=0\).
  • If \(deg(D(x)) = deg(N(x))\), then \(f(x)\) has a horizontal asymptote at \(y = \frac{a}{b}\), where \(a\) is the lead coefficient of \(N(x)\) and \(b\) is the lead coefficient of \(D(x)\).
  • If \(deg(N(x)) = deg(D(x)) + 1\), then \(f(x)\) has a slant (oblique) asymptote whose equation is the quotient after dividing \(N(x)\) by \(D(x)\) (ignoring the remainder).

Common Mistakes

• Incorrectly Identifying Domain Before Simplification: Finding the domain of a rational function after simplifying it by canceling common factors. The domain must be determined from the original, unsimplified function.
• Confusing Holes with Vertical Asymptotes: If a factor \((x-a)\) cancels from the numerator and denominator, there is a hole at \(x=a\), not a vertical asymptote, provided \(a\) is not still a zero of the simplified denominator.
• Errors in Determining Horizontal/Slant Asymptotes:
  • Making errors in calculating the ratio of leading coefficients for horizontal asymptotes when degrees are equal.
  • Errors in polynomial division when finding slant asymptotes.
• Assuming Graphs Cannot Cross Horizontal/Slant Asymptotes: While graphs of rational functions never cross their vertical asymptotes, they can cross their horizontal or slant asymptotes.
• Errors in Simplifying Difference Quotients for Rational Functions: Mistakes in finding a common denominator or in distributing terms when simplifying the complex fraction that arises.
• Solving Rational Inequalities Incorrectly: Making errors when finding critical points (zeros of numerator and denominator) or when testing intervals, or misinterpreting the graph if using the graphical method.