Rational Functions Review

Once again my procrastination (and inattentiveness to dates) leaves me writing up a test review the night before. Though I still doubt many of my classmates bothered studying until tonight.

Rational Functions

Definition: A polynomial over a polynomial.

Form: Where f(x) and g(x) are polynomials: [The function can also be written as the roots of f(x) over the roots for g(x).]

f(x)
g(x)

Domain: x belongs to real numbers except the zeros of the denominator: g(x) ≠ 0

Characteristics (examples for f(x) = 1/x²)

• intervals of increase or decrease
  • can be written in set notation:
    Increase {x E R x > 0}
    Decrease {x E R x <>
  • can be written in interval notation (remember, round brackets are "non-inclusive" and square brackets are "inclusive"):
    Increase [∞,0)
    Decrease (0,∞]
• x-intercepts
  • set the numerator equal to zero and solve for x. Check answers with denomenator to be sure x ≠ 0 (none for this function as 1≠0)
• y-intercept
  • solve for y when x = 0 (none for this function as the denominator ≠ 0)
• asymptotes (Definition: a limiting line; a line the function apporaches but does not reach)
  • Vertical (cannot be crossed): set denomenator = 0 and solve (VA = 0 for this function)
  • Horizontal (can be crossed): "analyze as x aproaches infinity"
    • divide each term on top and bottom by the highest degree of x
      f(x) = (1/x²)/1
    • since x is approaching infinity, anything over x will be a tiny number therefore as x --> infinity, f(x) --> 0/1
    • The HA for this function is 0.
    • check for crossing by setting HA = f(x) and solving.
  • Slant: Divide the numerator by the denomenator by long division. The resulting line is (potentially) your slant asymptote.
• domain and range
  • state in set notation
  • use asymptites as guides