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Pre-Calculus

Pre-Calculus. 2.6 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.

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Pre-Calculus

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  1. Pre-Calculus 2.6 Rational Functions

  2. 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

  3. Example 1 • Find the domain of f(x) = 5x/(x-1) and discuss the behavior of f(x) near any excluded values from the left and right.

  4. Asymptotes

  5. Vertical Asymptotes Vertical asymptotes – a line x=a such that A graph of f has vertical asymptotes at the zeros of D(x).

  6. Horizontal asymptotes – a line y=b such that • 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.

  7. Example 2 • Find all horizontal and vertical asymptotes. • A.

  8. 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.

  9. Example 3 • Find the slant asymptote of f(x) = x^3 • 2x^2-8

  10. Graphing Rational Functions • This is another area of mathematics in which our calculators can have trouble . • Often with vertical asymptotes, the calculator will graph something that is not part of the function.

  11. practice • Page 193: 1-79 odds

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