Rational functions are mathematical expressions that represent the ratio of two polynomials. They're crucial in modeling real-world scenarios like rates, proportions, and inverse variations. Understanding their behavior is key to solving complex problems in various fields.

Graphing rational functions involves identifying key features like asymptotes, intercepts, and end behavior. These characteristics help visualize the function's behavior and provide insights into its domain and range. Mastering rational functions opens doors to advanced mathematical concepts and practical applications.

Rational Function Notation and Domain

Arrow notation for rational functions

• Represents rational functions using arrow notation $f: X \to Y$
  • $X$ represents the domain or input values
  • $Y$ represents the codomain or possible output values
• Domain of a rational function includes all real numbers except values causing the denominator to equal zero
  • Values making the denominator zero are excluded from the domain

Domain restrictions of rational functions

• Find the domain by setting the denominator equal to zero and solving for x
  • Solutions are x-values not part of the domain
• Example: For $f(x) = \frac{x+1}{x-2}$, set $x-2=0$
  • Solving gives $x=2$
  • Domain is all real numbers except $x=2$, written as $\{x \in \mathbb{R} \mid x \neq 2\}$

Rational Function Applications and Graphing

Real-world applications of rational functions

• Model various real-world situations
  • Rates (speed, concentration)
  • Proportions (scale, similarity)
  • Inverse variations (work, cost)
• Steps to solve real-world problems:
  1. Identify given information and unknown quantity
  2. Set up a rational function representing the relationship between quantities
  3. Solve the equation or evaluate the function to find the desired result

Graphing techniques for rational functions

• Steps to graph a rational function:
  1. Find the domain and any vertical asymptotes
  2. Determine x- and y-intercepts (if any)
  3. Identify horizontal asymptotes and end behavior
  4. Plot points and sketch the graph
• Vertical asymptotes occur at x-values excluded from the domain
• Horizontal asymptotes and end behavior depend on degrees of numerator and denominator:
  • Numerator degree < denominator degree: horizontal asymptote is y = 0
  • Equal degrees: horizontal asymptote is $y = \frac{a_n}{b_n}$ ($a_n$ and $b_n$ are leading coefficients)
  • Numerator degree greater by 1: no horizontal asymptote, end behavior is a slant asymptote
  • Numerator degree greater by 2+: no horizontal asymptote, end behavior is a polynomial function

Rational Function Asymptotes and Behavior

Asymptotes in rational functions

• Vertical asymptotes occur where denominator equals zero, but numerator does not
  • Function approaches positive or negative infinity as x approaches vertical asymptote from either side
• Horizontal asymptotes occur when numerator degree ≤ denominator degree
  • Function approaches horizontal asymptote as x approaches positive or negative infinity

Behavior analysis of rational functions

• Near a vertical asymptote:
  • Function increases or decreases without bound (approaches infinity or negative infinity)
  • Function sign changes on either side of the asymptote
• Near a horizontal asymptote:
  • Function approaches asymptote y-value as x approaches positive or negative infinity
  • Function may approach asymptote from above or below
• As x approaches infinity:
  • End behavior depends on relative degrees of numerator and denominator (see graphing rules)
  • Function may approach a horizontal asymptote, slant asymptote, or polynomial function

Discontinuities and Special Cases

Types of discontinuities in rational functions

• Vertical asymptotes: occur when denominator equals zero and numerator doesn't
• Holes (removable discontinuities): occur when both numerator and denominator equal zero at the same x-value
  • Can be identified through factoring both numerator and denominator
• Jump discontinuities: occur in piecewise rational functions

Special rational functions

• Reciprocal functions: rational functions where numerator is a constant and denominator is a variable expression
  • Example: $f(x) = \frac{1}{x}$
• Polynomial functions: special case of rational functions where denominator is a constant
  • Have continuous graphs with no vertical asymptotes or holes