1. Section 3.7 Proper Rational Functions 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. 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 Rational function A function f(x) such that ) ) x x ( ( Q Q ) ) x x ( ( P P ) ) x x ( ( f f = = where P(x) and Q(x) are polynomials and Q(x) ≠ 0. DefinitionDefinition

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

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

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

8. EXAMPLE 1 Evaluate 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} 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 1 Evaluate f(x) and g(x) for x = 0 and x = 1 / 2. Give the domains. Let x 3 – 6x 2 + 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. x 3 – 6x 2 + 11x – 6 = 0 (x – 1)(x – 2)(x – 3) = 0 D = {x|x  1, 2, 3} Let x 3 – 6x 2 + 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. x 3 – 6x 2 + 11x – 6 = 0 (x – 1)(x – 2)(x – 3) = 0 D = {x|x  1, 2, 3}

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

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

12. EXAMPLE 2 Graph f(x) =. 1 x + 5 1 x + 5 x + 5 = 0 x =-5 D = {x|x ≠ -5} f(0) = =, plot (0, 1 / 5 ) There are no x-intercepts. x + 5 = 0 x =-5 D = {x|x ≠ -5} f(0) = =, plot (0, 1 / 5 ) There are no x-intercepts. 1 0 + 5 1 0 + 5 1515 1515

13. EXAMPLE 2 Graph f(x) =. 1 x + 5 1 x + 5 f(-6) = = = -1 1 -6 + 5 1 -6 + 5 1 1 f(-4) = = = 1 1 -4 + 5 1 -4 + 5 1111 1111

14. EXAMPLE 2 Graph f(x) =. 1 x + 5 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 ± . 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 3 Graph g(x) = x + 2 x 2 – 3x + 2 x + 2 x 2 – 3x + 2 x 2 – 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) x 2 – 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 3 Graph g(x) = x + 2 x 2 – 3x + 2 x + 2 x 2 – 3x + 2 g(3) = 5 / 2 = 2 1 / 2 ; plot (3, 2 1 / 2 ) g( 3 / 2 ) = -14; plot ( 3 / 2, -14) g(3) = 5 / 2 = 2 1 / 2 ; plot (3, 2 1 / 2 ) g( 3 / 2 ) = -14; plot ( 3 / 2, -14)

18. EXAMPLE 3 Graph g(x) = x + 2 x 2 – 3x + 2 x + 2 x 2 – 3x + 2 -224 -5 -10 -15

19. Homework: pp. 155-157 Homework: pp. 155-157

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

21. ►A. Exercises 1. ►A. Exercises 1.

22. ►A. Exercises 3. ►A. Exercises 3.

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

24. ►A. Exercises For each function give the y-intercept, the domain, and any vertical asymptotes. 7.h(x) = ►A. Exercises For each function give the y-intercept, the domain, and any vertical asymptotes. 7.h(x) = x 2 + 3x - 4 x 2 + 4x – 12 x 2 + 3x - 4 x 2 + 4x – 12

25. ►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) = ►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) = 6 x + 2 6 x + 2

26. ►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) = ►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) = x 2 + 3x – 4 x 2 + 4x – 12 x 2 + 3x – 4 x 2 + 4x – 12

27. ■ Cumulative Review Consider the following functions. f(x) =x 5 – x 3 k(x) =tan x g(x) =3x 2 + 1p(x) =x 6 – 4x h(x) =[x] q(x) =1/(x 2 – 4) j(x) =cos xr(x) =|x| ■ Cumulative Review Consider the following functions. f(x) =x 5 – x 3 k(x) =tan x g(x) =3x 2 + 1p(x) =x 6 – 4x h(x) =[x] q(x) =1/(x 2 – 4) j(x) =cos xr(x) =|x| 30. Which are odd functions?

28. ■ Cumulative Review Consider the following functions. f(x) =x 5 – x 3 k(x) =tan x g(x) =3x 2 + 1p(x) =x 6 – 4x h(x) =[x] q(x) =1/(x 2 – 4) j(x) =cos xr(x) =|x| ■ Cumulative Review Consider the following functions. f(x) =x 5 – x 3 k(x) =tan x g(x) =3x 2 + 1p(x) =x 6 – 4x h(x) =[x] q(x) =1/(x 2 – 4) j(x) =cos xr(x) =|x| 31. Which are even functions?

29. ■ Cumulative Review Consider the following functions. f(x) =x 5 – x 3 k(x) =tan x g(x) =3x 2 + 1p(x) =x 6 – 4x h(x) =[x] q(x) =1/(x 2 – 4) j(x) =cos xr(x) =|x| ■ Cumulative Review Consider the following functions. f(x) =x 5 – x 3 k(x) =tan x g(x) =3x 2 + 1p(x) =x 6 – 4x h(x) =[x] q(x) =1/(x 2 – 4) j(x) =cos xr(x) =|x| 32. Which are neither?

30. ■ Cumulative Review Let f(x) = 3x 5 – 23x 4 + 61x 3 – 61x 2 + 8x + 12. 33.List all possible integer zeros. ■ Cumulative Review Let f(x) = 3x 5 – 23x 4 + 61x 3 – 61x 2 + 8x + 12. 33.List all possible integer zeros.

31. ■ Cumulative Review Let f(x) = 3x 5 – 23x 4 + 61x 3 – 61x 2 + 8x + 12. 34.Factor f(x). ■ Cumulative Review Let f(x) = 3x 5 – 23x 4 + 61x 3 – 61x 2 + 8x + 12. 34.Factor f(x).