Algebra Rational Functions

Introduction  Rational Function – can be written in the form f(x) = N(x)/D(x)  N(x) and D(x) are polynomials with no common factors, D(x) is not zero  As usual, the domain of f(x) includes all values such that D(x) is not equal to zero

Example 1  Find the domain of  A. f(x) = 5x/(x-1)  B. g(x)=1/(x 2 - 4)  C. h(x) = x 4 /(x 2 +4)

Asymptotes  Vertical asymptotes – a line x=a such that as x approaches a, the function approaches positive or negative infinity  A graph of f has vertical asymptotes at the zeros of D(x).

 Horizontal asymptotes – a line y=b such that as x approaches positive or negative infinity, the function approaches b

 Horizontal asymptotes are found by comparing the degrees of N(x) and D(x).  If N is less in degree than D, then y=0 is a horizontal asymptote.  If N is the same degree as D then, the line y=p/q (p and q are the leading coefficients) is a horizontal asymptote.  If N is greater in degree than D, then there is no HA.

Example 2  Find all horizontal and vertical symptotes.  A. B.

Oblique (Slant) Asymptotes  If the degree of N is exactly one more than the degree of D, the function has a slant (or oblique) asymptote.  Use long division (or synthetic when possible) to find the asymptote.

Example 3  Find the slant asymptote of f(x) =

Graphing Rational Functions  Often with vertical asymptotes, the calculator will graph something that is not part of the function.

 Page 343: odds practice