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Properties of Rational Functions

Properties of Rational Functions. Learning Objectives. 1. Find the domain of a rational function 2. Find the vertical asymptotes of a rational function 3. Find the horizontal or oblique asymptotes of a rational function. Rational Function. To Find the Domain. Domain

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Properties of Rational Functions

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  1. Properties of Rational Functions

  2. Learning Objectives 1. Find the domain of a rational function 2. Find the vertical asymptotes of a rational function 3. Find the horizontal or oblique asymptotes of a rational function

  3. Rational Function

  4. To Find the Domain Domain The domain of a rational function is all real values except where the denominator, q(x) = 0

  5. Example

  6. Example

  7. Example

  8. Points Not in The Domain

  9. Holes and Vertical Asymptotes

  10. Examples Find holes and vertical asymptotes

  11. Examples Find holes and vertical asymptotes

  12. Example Find holes and vertical asymptotes

  13. Example

  14. More on Holes

  15. Holes and Vertical Asymptotes Holes and vertical asymptotes are discontinuities, but they are very different vertical asymptotes are non-removable discontinuities but holes are removable discontinuities, and by the addition of a point, we can create a function continuous at that point

  16. Example Hole at x = 2 X-2 evenly divides both the numerator and the denominator Holes do not appear on the graph, but are clearly indicated on the table

  17. Example Vertical Asymptote at x = 2 Holes do appear on the graph and are clearly indicated on the table

  18. Example x  0- f(x)  ∞ x  0+ f(x)  ∞ x  0 f(x)  ∞

  19. Horizontal and Oblique Asymptotes

  20. End Behavior A function will not have both an oblique and a horizontal asymptote

  21. Horizontal Asymptote A horizontal line is an asymptote only to the far left and the far right of the graph. "Far" left or "far" right is defined as anything past the vertical asymptotes or x-intercepts. Horizontal asymptotes are not asymptotic in the middle. It is okay to cross a horizontal asymptote in the middle.

  22. Example Find equation for horizontal asymptote

  23. Example Find equation for horizontal asymptote

  24. Example Find x-value(s) where f(x) crosses horizontal asymptote

  25. Example Find equation for horizontal asymptote

  26. Example Find equation for horizontal asymptote

  27. Example Find equation for horizontal asymptote

  28. Example

  29. Example This means for very large values of R2 the total resistance approaches 10 ohms.

  30. Oblique Asymptotes When the degree of the numerator is exactly one more than the degree of the denominator, the graph of the rational function will have an oblique asymptote. Another name for an oblique asymptote is a slant asymptote. To find the equation of the oblique asymptote, perform long division (synthetic if it will work) by dividing the denominator into the numerator and discarding the remainder

  31. Example Oblique Asymptote y = x+2 X-2 divides the numerator with a remainder Y2 is the end behavior of y1

  32. Finding Oblique Asymptotes

  33. Example

  34. Example

  35. Example

  36. Example

  37. Example Using synthetic division

  38. Example Our rational function Our rational function and OA

  39. Example

  40. Example Using synthetic division

  41. Example Our rational function Our rational function and OA

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