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 rationalfunctions.

3. To graph reduced proper rationalfunctions.

Objectives:1. To identify proper rational

functions in reduced form.2. To identify horizontal and vertical

asymptotes, domains, and ranges of reduced proper rationalfunctions.

3. To graph reduced proper rationalfunctions.

Rational function A function

f(x) such that

Rational function A function

f(x) such that ))xx((QQ))xx((PP

))xx((ff ==

where P(x) and Q(x) are polynomials and Q(x) ≠ 0.where P(x) and Q(x) are polynomials and Q(x) ≠ 0.

DefinitionDefinitionDefinitionDefinition

Examples of rational functions are

f(x) = and

g(x) = .

Examples of rational functions are

f(x) = and

g(x) = .

x2 – 5x + 1x2 – 5x + 1

3x + 2x3 – 6x2 + 11x - 6

3x + 2x3 – 6x2 + 11x - 6

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.

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.

EXAMPLE 1 Evaluate f(x) and g(x) for x = 0 and x = 1/2. Give the domains.EXAMPLE 1 Evaluate f(x) and g(x) for x = 0 and x = 1/2. Give the domains.

f(x) = f(x) = x2 – 5x + 1x2 – 5x + 1

f(0) = = = -5f(0) = = = -502 – 50 + 102 – 50 + 1

-51-51

f(1/2) = = = -19/6f(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

3(0) + 203 – 6(0)2 + 11(0) - 6

3(0) + 203 – 6(0)2 + 11(0) - 6

g(0) = = 2/-6 = -1/3g(0) = = 2/-6 = -1/3

EXAMPLE 1 Evaluate f(x) and g(x) for x = 0 and x = 1/2. Give the domains.EXAMPLE 1 Evaluate f(x) and g(x) for x = 0 and x = 1/2. Give the domains.

3x + 2x3 – 6x2 + 11x - 6

3x + 2x3 – 6x2 + 11x - 6

g(x) = 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

EXAMPLE 1 Evaluate f(x) and g(x) for x = 0 and x = 1/2. Give the domains.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}

EXAMPLE 1 Evaluate f(x) and g(x) for x = 0 and x = 1/2. Give the domains.EXAMPLE 1 Evaluate 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) = 0D = {x|x 1, 2, 3}

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) = 0D = {x|x 1, 2, 3}

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

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

DefinitionDefinitionDefinitionDefinition

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

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

EXAMPLE 2 Graph f(x) = .EXAMPLE 2 Graph f(x) = .1

x + 51

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.

10 + 5

10 + 5

1515

EXAMPLE 2 Graph f(x) = .EXAMPLE 2 Graph f(x) = .1

x + 51

x + 5

f(-6) = = = -1f(-6) = = = -11

-6 + 51

-6 + 51-11-1

f(-4) = = = 1f(-4) = = = 11

-4 + 51

-4 + 51111

EXAMPLE 2 Graph f(x) = .EXAMPLE 2 Graph f(x) = .1

x + 51

x + 5

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 ±.

EXAMPLE 3 Graph g(x) = EXAMPLE 3 Graph g(x) = x + 2

x2 – 3x + 2x + 2

x2 – 3x + 2

x2 – 3x + 2 = 0(x – 2)(x – 1) = 0x = 2 or x = 1

g(0) = 2/2 = 1; plot (0, 1)

x + 2 = 0 at x = -2; (-2, 0)

x2 – 3x + 2 = 0(x – 2)(x – 1) = 0x = 2 or x = 1

g(0) = 2/2 = 1; plot (0, 1)

x + 2 = 0 at x = -2; (-2, 0)

EXAMPLE 3 Graph g(x) = EXAMPLE 3 Graph g(x) = x + 2

x2 – 3x + 2x + 2

x2 – 3x + 2

g(3) = 5/2 = 21/2; plot (3, 21/2)

g(3/2) = -14; plot (3/2, -14)

g(3) = 5/2 = 21/2; plot (3, 21/2)

g(3/2) = -14; plot (3/2, -14)

EXAMPLE 3 Graph g(x) = EXAMPLE 3 Graph g(x) = x + 2

x2 – 3x + 2x + 2

x2 – 3x + 2

-2 2 4

-5

-10

-15

Homework:

pp. 155-157

Homework:

pp. 155-157

►A. ExercisesFor each graph below, identify the intercepts, asymptotes, domain, and range.

►A. ExercisesFor each graph below, identify the intercepts, asymptotes, domain, and range.

►A. Exercises1.

►A. Exercises1.

►A. Exercises3.

►A. Exercises3.

►A. ExercisesFor each function give the y-intercept, the domain, and any vertical asymptotes.

5. f(x) =

►A. ExercisesFor each function give the y-intercept, the domain, and any vertical asymptotes.

5. f(x) = 6

x + 26

x + 2

►A. ExercisesFor each function give the y-intercept, the domain, and any vertical asymptotes.

7. h(x) =

►A. ExercisesFor each function give the y-intercept, the domain, and any vertical asymptotes.

7. h(x) = x2 + 3x - 4

x2 + 4x – 12x2 + 3x - 4

x2 + 4x – 12

►B. ExercisesDecide 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. ExercisesDecide 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 + 26

x + 2

►B. ExercisesDecide 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. ExercisesDecide 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) =x2 + 3x – 4

x2 + 4x – 12x2 + 3x – 4

x2 + 4x – 12

■ Cumulative ReviewConsider the following functions.

f(x) = x5 – x3 k(x) = tan xg(x) = 3x2 + 1 p(x) = x6 – 4xh(x) = [x] q(x) = 1/(x2 – 4)j(x) = cos x r(x) = |x|

■ Cumulative ReviewConsider the following functions.

f(x) = x5 – x3 k(x) = tan xg(x) = 3x2 + 1 p(x) = x6 – 4xh(x) = [x] q(x) = 1/(x2 – 4)j(x) = cos x r(x) = |x|

30. Which are odd functions?30. Which are odd functions?

■ Cumulative ReviewConsider the following functions.

f(x) = x5 – x3 k(x) = tan xg(x) = 3x2 + 1 p(x) = x6 – 4xh(x) = [x] q(x) = 1/(x2 – 4)j(x) = cos x r(x) = |x|

■ Cumulative ReviewConsider the following functions.

f(x) = x5 – x3 k(x) = tan xg(x) = 3x2 + 1 p(x) = x6 – 4xh(x) = [x] q(x) = 1/(x2 – 4)j(x) = cos x r(x) = |x|

31. Which are even functions?31. Which are even functions?

■ Cumulative ReviewConsider the following functions.

f(x) = x5 – x3 k(x) = tan xg(x) = 3x2 + 1 p(x) = x6 – 4xh(x) = [x] q(x) = 1/(x2 – 4)j(x) = cos x r(x) = |x|

■ Cumulative ReviewConsider the following functions.

f(x) = x5 – x3 k(x) = tan xg(x) = 3x2 + 1 p(x) = x6 – 4xh(x) = [x] q(x) = 1/(x2 – 4)j(x) = cos x r(x) = |x|

32. Which are neither?32. Which are neither?

■ Cumulative ReviewLet f(x) = 3x5 – 23x4 + 61x3 – 61x2 + 8x + 12.33. List all possible integer zeros.

■ Cumulative ReviewLet f(x) = 3x5 – 23x4 + 61x3 – 61x2 + 8x + 12.33. List all possible integer zeros.

■ Cumulative ReviewLet f(x) = 3x5 – 23x4 + 61x3 – 61x2 + 8x + 12.34. Factor f(x).

■ Cumulative ReviewLet f(x) = 3x5 – 23x4 + 61x3 – 61x2 + 8x + 12.34. Factor f(x).