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Section 3.7 Rational Functions

Chapter 3 – Polynomial and Rational Functions. Section 3.7 Rational Functions. Example. Rational functions are quotients of polynomials. For example, functions that can be expressed as where P ( x ) and Q ( x ) are polynomials and Q ( x )  0 .

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Section 3.7 Rational Functions

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  1. Chapter 3 – Polynomial and Rational Functions Section 3.7 Rational Functions 3.7 - Rational Functions

  2. Example • Rational functions are quotients of polynomials. For example, functions that can be expressed as where P(x) and Q(x) are polynomials and Q(x)  0. Note: We assume that P(x) and Q(x) have no factors in common. 3.7 - Rational Functions

  3. Basic Rational Function We want to identify the characteristics of rational functions 3.7 - Rational Functions

  4. Domain In order to find the domain of a rational function, we must set the denominator equal to zero. These values are where our function does not exist. Hint: If possible, always factor the denominator first before finding the domain. 3.7 - Rational Functions

  5. Arrow Notation • We will be using the following arrow notation for asymptotes: 3.7 - Rational Functions

  6. Vertical Asymptotes • The line x = a is a vertical asymptote of the function y = f (x) if y approaches ∞ as x approaches a from the right or left. 3.7 - Rational Functions

  7. Vertical Asymptotes (VA) To find the VA 1. Set the denominator = 0 and solve for x. 2. Check using arrow notation. 3.7 - Rational Functions

  8. Horizontal Asymptotes • The line y= bis a horizontal asymptote of the function y = f (x) if yapproaches b as xapproaches ∞. 3.7 - Rational Functions

  9. Horizontal Asymptotes (HA) • To find the HA, we let r be the rational function 1. If n < m, then r has the horizontal asymptote y=0. 2. If n =m, then r has the horizontal asymptote . 3. If n> m, then r has no horizontal asymptotes. We need to check for a slant asymptote (SA). 3.7 - Rational Functions

  10. Slant Asymptotes • To find the SA, we perform long division and get where R(x)/Q(x) is the remainder and the SA is y = ax + b. 3.7 - Rational Functions

  11. Example Given the above equation, find the characteristics of rational functions and sketch a graph of the function. 3.7 - Rational Functions

  12. Example Given the above equation, find the characteristics of rational functions and sketch a graph of the function. 3.7 - Rational Functions

  13. Example Given the above equation, find the characteristics of rational functions and sketch a graph of the function. 3.7 - Rational Functions

  14. Example Given the above equation, find the characteristics of rational functions and sketch a graph of the function. 3.7 - Rational Functions

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