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Section 3.7 Proper Rational Functions

Section 3.7 Proper Rational Functions. Objectives: 1. To identify proper rational functions in reduced form. 2. To identify horizontal and vertical asymptotes, domains, and ranges of reduced proper rational functions. 3. To graph reduced proper rational functions.

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Section 3.7 Proper Rational Functions

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  1. Section 3.7 Proper Rational Functions

  2. Objectives: 1. To identify proper rational functions in reduced form. 2. To identify horizontal and vertical asymptotes, domains, and ranges of reduced proper rational functions. 3. To graph reduced proper rational functions.

  3. Rational function A function f(x) such that P ( x ) f ( x ) = Q ( x ) where P(x) and Q(x) are polynomials and Q(x) ≠ 0. Definition

  4. x2 – 5 x + 1 3x + 2 x3 – 6x2 + 11x - 6 Examples of rational functions are f(x) = and g(x) = .

  5. As with reciprocal functions, you can find the domain by excluding values where the denominator is zero. To evaluate a rational function, substitute the given domain value and simplify to find the range value.

  6. f(x) = x2 – 5 x + 1 02 – 5 0 + 1 -5 1 f(0) = = = -5 -19/4 3/2 f(1/2) = = = -19/6 (1/2)2 – 5 1/2 + 1 EXAMPLE 1Evaluate f(x) and g(x) for x = 0 and x = 1/2. Give the domains.

  7. g(x) = 3x + 2 x3 – 6x2 + 11x - 6 3(0) + 2 03 – 6(0)2 + 11(0) - 6 7/2 -15/8 g(0) = = 2/-6 = -1/3 3(1/2) + 2 (1/2)3 – 6(1/2)2 + 11(1/2) - 6 g(1/2) = = = -28/15 EXAMPLE 1Evaluate f(x) and g(x) for x = 0 and x = 1/2. Give the domains.

  8. EXAMPLE 1Evaluate f(x) and g(x) for x = 0 and x = 1/2. Give the domains. Let x + 1 = 0 to determine when the denominator of f(x) will be 0. x + 1 = 0 x = -1 D = {x|x  -1}

  9. EXAMPLE 1Evaluate f(x) and g(x) for x = 0 and x = 1/2. Give the domains. Let x3 – 6x2 + 11x – 6 = 0 to determine when the denominator of g(x) will be 0. Possible rational zeros are ±1, ±2, ±3, ±6. Use synthetic division to factor the polynomial. x3 – 6x2 + 11x – 6 = 0 (x – 1)(x – 2)(x – 3) = 0 D = {x|x  1, 2, 3}

  10. Definition Proper rational function A rational function in which the degree of the numerator is less than the degree of the denominator.

  11. Areduced rational function is a function in which the numerator and denominator have no common factors.

  12. EXAMPLE 2Graph f(x) = . 1 x + 5 1 0 + 5 1 5 x + 5 = 0 x = -5 D = {x|x ≠ -5} f(0) = = , plot (0, 1/5) There are no x-intercepts.

  13. EXAMPLE 2Graph f(x) = . 1 x + 5 f(-6) = = = -1 1 1 f(-4) = = = 1 1 -1 1 -6 + 5 1 -4 + 5

  14. EXAMPLE 2Graph f(x) = . 1 x + 5

  15. Vertical asymptotes occur at any x-value that makes the denominator of a reduced rational function equal to zero. Horizontal asymptotes is y = 0 for every proper rational function. Horizontal asymptotes tell what happens as x approaches ±.

  16. EXAMPLE 3Graph g(x) = x + 2 x2 – 3x + 2 x2 – 3x + 2 = 0 (x – 2)(x – 1) = 0 x = 2 or x = 1 g(0) = 2/2 = 1; plot (0, 1) x + 2 = 0 at x = -2; (-2, 0)

  17. EXAMPLE 3Graph g(x) = x + 2 x2 – 3x + 2 g(3) = 5/2 = 21/2; plot (3, 21/2) g(3/2) = -14; plot (3/2, -14)

  18. EXAMPLE 3Graph g(x) = x + 2 x2 – 3x + 2 -2 2 4 -5 -10 -15

  19. Homework: pp. 155-157

  20. ►A. Exercises For each graph below, identify the intercepts, asymptotes, domain, and range.

  21. ►A. Exercises 1.

  22. ►A. Exercises 3.

  23. 6 x + 2 ►A. Exercises For each function give the y-intercept, the domain, and any vertical asymptotes. 5. f(x) =

  24. x2 + 3x - 4 x2 + 4x – 12 ►A. Exercises For each function give the y-intercept, the domain, and any vertical asymptotes. 7. h(x) =

  25. 6 x + 2 ►B. Exercises Decide if each function above is proper and reduced. If not, explain why. If it is, give the x-intercept and the horizontal asymptotes. Graph each function and identify any graphs that are continuous, odd, or even. 11. f(x) =

  26. x2 + 3x – 4 x2 + 4x – 12 ►B. Exercises Decide if each function above is proper and reduced. If not, explain why. If it is, give the x-intercept and the horizontal asymptotes. Graph each function and identify any graphs that are continuous, odd, or even. 13. h(x) =

  27. ■ Cumulative Review Consider the following functions. f(x) = x5 – x3 k(x) = tan x g(x) = 3x2 + 1 p(x) = x6 – 4x h(x) = [x] q(x) = 1/(x2 – 4) j(x) = cos x r(x) = |x| 30. Which are odd functions?

  28. ■ Cumulative Review Consider the following functions. f(x) = x5 – x3 k(x) = tan x g(x) = 3x2 + 1 p(x) = x6 – 4x h(x) = [x] q(x) = 1/(x2 – 4) j(x) = cos x r(x) = |x| 31. Which are even functions?

  29. ■ Cumulative Review Consider the following functions. f(x) = x5 – x3 k(x) = tan x g(x) = 3x2 + 1 p(x) = x6 – 4x h(x) = [x] q(x) = 1/(x2 – 4) j(x) = cos x r(x) = |x| 32. Which are neither?

  30. ■ Cumulative Review Let f(x) = 3x5 – 23x4 + 61x3 – 61x2 + 8x + 12. 33. List all possible integer zeros.

  31. ■ Cumulative Review Let f(x) = 3x5 – 23x4 + 61x3 – 61x2 + 8x + 12. 34. Factor f(x).

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