SECTION 5.1 Simplifying Rational Expressions
Chapter 5
Rational Expressions, Equations, and Functions
Section 5.1:
Simplifying Rational Expressions
 Rational Expressions
Rational Expressions
Definition:
Simplifying:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 5 Rational Expressions, Equations, and Functions
Example:
Solution:
298
University of Houston Department of Mathematics
SECTION 5.1 Simplifying Rational Expressions
Additional Example 1:
Solution:
Additional Example 2:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 3:
Solution:
Additional Example 4:
300
University of Houston Department of Mathematics
SECTION 5.1 Simplifying Rational Expressions
Solution:
MATH 1300 Fundamentals of Mathematics
301
Exercise Set 5.1: Simplifying Rational Expressions
16.
x3
x2  9
17.
a 2  b2
ab
18.
x 2  16
x4
19.
49  c 2
c 2  9c  18
20.
x 2  11x  10
100  x 2
21.
x 2  2 x  15
x 2  10 x  21
22.
m2  m  20
m2  m  30
23.
x2  5x  6
x 2  x  12
24.
x 2  7 x  12
x 2  7 x  30
x y
yx
25.
x 2  8 x  12
x 2  13x  42
cd
10.
d c
26.
x 2  7 x  10
x 2  7 x  10
27.
x 2  36
x 2  12 x  36
28.
x2  8x  16
x2  16
29.
9 x  36
x2  4 x
7 x 2  14 x
x2
10 x 2  30 x
5 x 2  10 x
Simplify the following rational expressions. If the
expression cannot be simplified any further, then
simply rewrite the original expression.
1.
15
25
2.

30
36
3.

48
64
4.
26
39
5.
6.
7.
8.
9.
11.
60 x 2 y 5
48 x 5 y 3


49a 4 b9
56a 7 b10
5 x3  x  y 
10 x5  x  y 
8c 6  c  d 
2
12c  c  d 
3
3
2  a  b  c  d 
6 b  a 
12. 
12  x  y  w  z 
6  z  w x  y 
4x  8
13.
x2
302
7
14.
x3
5 x  15
30.
15.
x5
x 2  25
31.
University of Houston Department of Mathematics
Exercise Set 5.1: Simplifying Rational Expressions
32.
6 x2  8x
9 x3  12 x 2
48.
x5
x  125
33.
x2  7 x  6
8x2  8x
49.
x3  27
x 2  3x  9
34.
4 x 2  20 x
x2  4 x  5
50.
x3  1
x2  x  1
35.
6 x 2  24 x  18
4 x 2  8 x  60
36.
5x2  10 x  40
10 x2  30 x  20
37.
4 x 2  17 x  15
5 x 2  13x  6
38.
4 x 2  8 x  21
8 x 2  24 x  14
39.
6 x2  5x  4
10 x 2  9 x  2
40.
15x 2  4 x  4
5x2  22 x  8
41.
8x2  30 x  7
16 x 2  1
42.
9 x 2  25
6 x 2  13x  5
43.
m3  m2  m  1
m3  m  m2 n  n
44.
ax  ay  bx  by
ax  ay  2 x  2 y
45.
xy  3x  2 y  6
yz  3z  5 y  15
46.
ab  5a  2b  10
a 2 b  4b  5a 2  20
47.
x3  8
x2
MATH 1300 Fundamentals of Mathematics
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303
CHAPTER 5 Rational Expressions, Equations, and Functions
Section 5.2: Multiplying and Dividing Rational Expressions
 Multiplication and Division
Multiplication and Division
Multiplication of Rational Expressions:
To multiply two fractions, place the product of the numerators over the product
of the denominators.
Example:
Solution:
304
University of Houston Department of Mathematics
SECTION 5.2 Multiplying and Dividing Rational Expressions
Division of Rational Expressions:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 1:
Solution:
Additional Example 2:
306
University of Houston Department of Mathematics
SECTION 5.2 Multiplying and Dividing Rational Expressions
Solution:
Additional Example 3:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 5 Rational Expressions, Equations, and Functions
Solution:
308
University of Houston Department of Mathematics
Exercise Set 5.2: Multiplying and Dividing Rational Expressions
Multiply the following rational expressions and
simplify. No answers should contain negative
exponents.
16. ( x  2) 
3
2 x
1.
6 14

7 18
17. ( x  5) 
3x  2
5 x
2.
8 45
 
9 32
18. (3  x) 
2x 1
x3
3.
10 
2
5
19. ( x  4) 
3
5 x  20
4.
12 
5.
ab4 c7 d 3

c 5 d 8 a 6 b9
21. (2 x  8) 
3x  4
3 x  12
6.
x5 y 6 wz 3

w3 z 8 x10 y 9
22. (3 x  3) 
2x  2
4x  4
23.
7.
 m8 n 2   n 4t 6   p 6t 2 
  3 5    5    3 7 
 pt   m   mn 
6 x  12 4 x  12

x  3 3x  6
x  7 6 x  24

2 x  8 5 x  35
8.
x y   a b
 2   7 7
 ab   x y
24.
25.
6 x  10
3

5 x 15  9 x
26.
2x 4x  6

6 x  9 x2  x
27.
x2  x  6 x2  6 x  5

x 2  3x  4 x 2  2 x  15
28.
x 2  x  2 x 2  x  12

x 2  8x  15 x 2  9 x  14
29.
x 2  3x  10 2 x 2  4 x

x5
6 x2  24 x
30.
6 x 2  30 x x 2  4 x  21

x 2  x  6 40 x  8x 2
31.
x  4 x2  9

3  x x 2  16
20. (4 x  28) 
3
8
3 4
9.
2 x 2 
10. 6 x 4 
11.
12.
13.
8 2
  x y a 
  5

b y 
 
2 3 4
3
6 x5
5
2x
x5 x3

x  3 x  10
x  6 5

x 1 x  6
 x  2 
14. ( x  1) 
15. (7  x) 
x5
x2
x3
x 1
5 x
x7
MATH 1300 Fundamentals of Mathematics
2
x7
309
Exercise Set 5.2: Multiplying and Dividing Rational Expressions
32.
x 2  25 x 2  12 x  36

x6
x5
47.
x3 x5

x 1 x 1
33.
2 x 2  9 x  10 x 2  7 x  12

x 2  5 x  6 2 x 2  3x  5
48.
x 4 x3

x2 x2
34.
x 2  2 x  8 3x 2  14 x  5

3x 2  16 x  5 x 2  x  20
49.
7 x  7 x2  1

21x
3x
35.
ax  bx  ay  by ax  7 x  2a  14

ax  7 x  3a  21 ax  bx  2a  2b
50.
7
x4

x2  9 x  3
36.
ac  2ad  bc  2bd
c2  d 2

ac  ad  bc  bd 3ac  3ad  bc  bd
51.
x2  1 x  1

x  6 3x  18
52.
x
5x

x2  4 x  2
5 15

37.
8 32
53.
5
10

2
x4
16  x
6 4

38.
25 5
54.
x2  4 2  x

x  5 25  x 2
 25 
39. 10    
 2 
55.
x2  9
x3

x2  1 x2  2 x  1
12
40. 6 
7
56.
4 x2  9
2x  3

x  10 x  25 x  5
2
  6 
41.
3
57.
x 2  3x  10 x 2  x  6

x 2  3x  28 x2  x  12
 4
42.      40 
 5
58.
x 2  4 x  4 x 2  8 x  20

x 2  6 x  16 x 2  9 x  8
x
x4 z3
43. 2 7  5
y z
y
59.
6 x 2  x  1 3x 2  2 x  1

6 x 2  5 x  1 3x 2  4 x  1
a 3 c 7 b5 c 9
44.
 2
b4
a
60.
10 x 2  17 x  6 6 x 2  5 x  4

5x 2  4 x  12 3x 2  2 x  8
a 5b6
45. 2 5  a5 d 2
c d
61.
am  an  bm  bn am  an  3bm  3bn

am  an  bm  bn am  an  3bm  3bn
x3 y 2
46. x z  6
w z
62.
cx  2dx  cy  2dy cx  cy  5dx  5dy

cx  5dx  c  5d
x 2  x  3xy  3 y
Divide the following rational expressions and simplify.
No answers should contain negative exponents.
4 5
310
2
University of Houston Department of Mathematics
SECTION 5.3 Adding and Subtracting Rational Expressions
Section 5.3:
Adding and Subtracting Rational Expressions
 Addition and Subtraction
Addition and Subtraction
Addition and Subtraction of Rational Expressions with Like
Denominators:
Example:
Perform the following operations. All results should be in simplified form.
MATH 1300 Fundamentals of Mathematics
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CHAPTER 5 Rational Expressions, Equations, and Functions
Solution:
Addition and Subtraction of Rational Expressions with Unlike
Denominators:
Example:
312
University of Houston Department of Mathematics
SECTION 5.3 Adding and Subtracting Rational Expressions
Solution:
Additional Example 1:
Perform the following operations. All results should be in simplified form.
MATH 1300 Fundamentals of Mathematics
313
CHAPTER 5 Rational Expressions, Equations, and Functions
Solution:
Additional Example 2:
Perform the addition. Give the result in simplified form.
Solution:
314
University of Houston Department of Mathematics
SECTION 5.3 Adding and Subtracting Rational Expressions
Additional Example 3:
Perform the subtraction. Give the result in simplified form.
Solution:
MATH 1300 Fundamentals of Mathematics
315
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 4:
Perform the subtraction. Give the result in simplified form.
Solution:
316
University of Houston Department of Mathematics
SECTION 5.3 Adding and Subtracting Rational Expressions
Additional Example 5:
Perform the following operations. Give all results in simplified form.
MATH 1300 Fundamentals of Mathematics
317
CHAPTER 5 Rational Expressions, Equations, and Functions
Solution:
318
University of Houston Department of Mathematics
Exercise Set 5.3: Adding and Subtracting Rational Expressions
Perform the indicated operations and simplify.
(Whenever possible, write both the numerator and
denominator of the answer in factored form.)
16.
1
2

x2 x2
1.
2x 3y

5
7
17.
6
2

x3 x7
2.
4a 2b

5
3
18.
7
4

x9 x2
3.
3
2

4a 9b
19.
5
x4

x 3 3 x
4.
7
5

2c 3c
20.
x
2 x

5 x x5
5.
7
2
 5
2 2
x y
xy
21. 3 
2
x5
6.
3
2

a 4 b7 a 5b 4
22. 4 
5
x7
7.
x8 x7

x5 x5
23.
7
2
x2
8.
x  3 4x  6

x 1 x 1
24.
6
4
x3
9.
3x  2
2x  6

5 x  20 5 x  20
25.
x4
2

x 1 x  3
2 x  3 10 x  9

10.
4x  3 4x  3
26.
x3
1

x3 x3
2
3

11.
x 1 x  5
27.
x2
3

x
x2
12.
5
6

x4 x7
28.
x
4

x3 x5
13.
3
8

x x 1
29.
x 1
1

x 1 2x 1
5
2

14.
x x4
30.
2x  3
6

x
x 1
3
4

15.
x 1 x  2
31.
x  3 x 1

x2 x4
MATH 1300 Fundamentals of Mathematics
319
Exercise Set 5.3: Adding and Subtracting Rational Expressions
32.
x 1 x  2

x  3 x 1
33.
x5 x2

x4 x3
34.
x4 x2

x 1 x 1
35.
7
5

8 x  12 6 x  6
36.
5
2

12 x  6 10 x  40
37.
3
8
6


x x 1 x  2
38.
2
4
3


x3 x2 x
39. 5 
320
4 x2  3
x2  2 x  8
40.
3x 2  5 x
2
x 2  3x  4
41.
x
2
5
 2
 2
x  2x  8 x  2x x  4x
42.
x
1
1
 2
 2
x  3x  18 x  6 x x  3x
43.
x
4
2


x 2  10 x  24 x 2  12 x  32 x 2  14 x  48
44.
x
2
3


x 2  7 x  12 x 2  4 x  3 x 2  5 x  4
2
2
University of Houston Department of Mathematics
SECTION 5.4 Complex Fractions
Section 5.4:
Complex Fractions
 Simplifying Complex Fractions
Simplifying Complex Fractions
Definition:
Simplifying:
MATH 1300 Fundamentals of Mathematics
321
CHAPTER 5 Rational Expressions, Equations, and Functions
Example:
Solution:
Method 1:
322
University of Houston Department of Mathematics
SECTION 5.4 Complex Fractions
Method 2:
MATH 1300 Fundamentals of Mathematics
323
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 1:
Solution:
Additional Example 2:
Solution:
324
University of Houston Department of Mathematics
SECTION 5.4 Complex Fractions
Additional Example 3:
Solution:
MATH 1300 Fundamentals of Mathematics
325
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 4:
Solution:
326
University of Houston Department of Mathematics
SECTION 5.4 Complex Fractions
Additional Example 5:
Solution:
MATH 1300 Fundamentals of Mathematics
327
Exercise Set 5.4: Complex Fractions
Simplify the following. No answers should contain
negative exponents.
1.
2.
3.
7
12
5
8
10.
11.
6
7
2
3
5.
6.
7.
8.
9.
328
5
6
1 2

2 3
1
12.
4x
y
x4
5 y3
13.
12a
b
4c 2
b7
2ab 2
c3 d
8a 3 c
5bd 4
3x 4 y
8 w6 z 5
9 x2 w
4z7
4 x3 yz
8x2
y5
5b3c
2d
10bd
2 3

3 4
1 2

2 5
14.
1
7
1
7
2
5
4.
5 1

6 2
3 2

5 3
1
4
1
2
2
5
8
4
2
3
15.
x2
5
x3
10
16.
x7
8
x 1
12
17.
ab
b
ba
ab 3
18.
19.
x2 y
x y
xy
x 2  7 x  12
8x
x 2  x  20
6x4
University of Houston Department of Mathematics
Exercise Set 5.4: Complex Fractions
20.
9 x5
x  6 x  16
18 x3
x 2  11x  24
21.
x x

3 2
x x

3 2
22.
a a

4 5
a a

3 4
23.
x2  1
x
1 1

2 2x
24.
5 1

3x 3
25  x 2
x
25.
2 3

a b
5 4

a b
26.
7 2

x y
3 4

x y
2
6
x 1
9
x
x
x
27.
4
x
10
x
x7
30.
6x
x  x2
4
1

x  2 x 1
31.
1
3

x3 x4
3
2

x  3 x 1
32.
2
5

x 1 x  2
3
2

x2 x
33.
15
x
12
x7
x
2
x2
34.
For each of the following expressions,
(a) Rewrite the expression so that it contains
positive exponents rather than negative
exponents.
(b) Simplify the expression.
35.
x1
x 1
36.
3  x 1
x 1
29.
2
3

x5 x5
2
2
x  25
MATH 1300 Fundamentals of Mathematics
1
37.
x  1  y 1
x 1  y  1
38.
c 1  d 1
c 2  d 2
39.
x 2  y 2
x 1  y 1
x
28.
14
x
7
x6
x
x9
329
Exercise Set 5.4: Complex Fractions
40.
a 1  b 1
b 2  a 2
41.
c 1  d 1
c 3  d 3
42.
x 3  y 3
x 1  y 1
43.
a 3  b 3
a 2  b 2
44.
x 2  y 2
x  3  y 3
45. 1 
1
1  x 1
46. 1 
1
1  x 2
47. 4 
48.
330
5
5  x 1
2
3
2  x 1
University of Houston Department of Mathematics
SECTION 5.5 Solving Rational Equations
Section 5.5:
Solving Rational Equations
 Rational Equations
Rational Equations
Definition of a Rational Equation:
Solving a Rational Equation:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
331
CHAPTER 5 Rational Expressions, Equations, and Functions
Example:
Solution:
332
University of Houston Department of Mathematics
SECTION 5.5 Solving Rational Equations
MATH 1300 Fundamentals of Mathematics
333
CHAPTER 5 Rational Expressions, Equations, and Functions
Extraneous Solutions:
Example:
Solution:
334
University of Houston Department of Mathematics
SECTION 5.5 Solving Rational Equations
Additional Example 1:
Solution:
MATH 1300 Fundamentals of Mathematics
335
CHAPTER 5 Rational Expressions, Equations, and Functions
Additional Example 2:
Solution:
336
University of Houston Department of Mathematics
SECTION 5.5 Solving Rational Equations
Additional Example 3:
Solution:
MATH 1300 Fundamentals of Mathematics
337
CHAPTER 5 Rational Expressions, Equations, and Functions
338
University of Houston Department of Mathematics
SECTION 5.5 Solving Rational Equations
Additional Example 4:
Solution:
MATH 1300 Fundamentals of Mathematics
339
Exercise Set 5.5: Solving Rational Equations
Solve the following. Remember to identify any
extraneous solutions.
3x  1
7
x5
1.
2x x
 2
5 3
18.
3x
 2
x7
2.
3a 2a

1
4
3
19.
7
1
x 9
3c 2c

 22
2 5
20.
5
0
x 4
5x x
  14
8 4
21.
x5
1
x  7 x  12
5 x 3x

 2
6 10
22.
11  x
1
x  3 x  10
6.
7 x 3x

5
8 20
23.
5
9

2
7 t 3
7.
4x  7 x  3

x5
x5
24.
3
12
 3
x 1 5
8.
3x  4 x  8

x2 x2
25.
7 x8

 1
9 x 1
9.
x  5 2x  6

x 1
x 1
26.
a2 1
  1
a9 4
10.
3x  4 5 x  7

x6
x6
27.
x7 2
 3
x7 3
11.
2
7

 3
5x 4 x
28.
2 t2

 2
3 t 3
7
5

2
6x 4x
29.
w  1 3 13
 
w  1 4 12
3
 4
x2
30.
x4 1 9
 
x  9 2 14
5
3
x7
31.
5 x3
7


3 x4 x4
2
0
x5
32.
x
3 x3
 
x2 7 x2
3.
4.
5.
12. 
13.
14.
15.
16.
340
17.
2
2
2
2
5x
0
x2
University of Houston Department of Mathematics
Exercise Set 5.5: Solving Rational Equations
33.
4
1
8
 
x  5 3 3x  15
49.
4
1

1
x  4 x 1
34.
7
4
5


x  2 3x  6
3
50.
5
2

1
x4 x2
35.
3
2
1


4a  8 3a  6 36
51.
7
8

1
x5 x8
36.
5
1
7


3c  15 2c  10 12
52.
5
6

1
x7 x9
37.
3
1
7


x  5 x  3 x 2  2 x  15
53.
x4
2

 1
x  5 x  10
38.
2
1
4


x  1 x  2 x2  x  2
54.
x2
1

 1
x7 x3
39.
4
2
8

 2
x  3 x  1 x  2x  3
55.
1
4
x


2 x  5 3x
2x  5
40.
7
2
10

 2
x  4 x  5 x  9 x  20
56.
2
1
6x
 
3x  1 x 3x  1
41.
3
4
8

 2
x2 x2 x 4
57.
4
3
3x


3x  2 x  1 x  1
42.
3
6
24

 2
x  4 x  4 x  16
58.
5
2
x
 
2x  3 x 2x  3
43. 1 
44.
1 6

x x2
12 1
 1
x2 x
45. 2 
7 4

x x2
46.
4 11
 3
x2 x
47.
6
1
 1
x4 x
48.
7
4

1
x x5
MATH 1300 Fundamentals of Mathematics
341
CHAPTER 5 Rational Expressions, Equations, and Functions
Section 5.6:
Rational Functions
 Working with Rational Functions
Working with Rational Functions
Definition of a Rational Function:
Domain of a Rational Function:
Example:
342
University of Houston Department of Mathematics
SECTION 5.6 Rational Functions
Solution:
MATH 1300 Fundamentals of Mathematics
343
CHAPTER 5 Rational Expressions, Equations, and Functions
Graph of a Rational Function:
Example:
Solution:
344
University of Houston Department of Mathematics
SECTION 5.6 Rational Functions
The graph of the function is shown below, labeled with the information from parts (b)-(d).
MATH 1300 Fundamentals of Mathematics
345
CHAPTER 5 Rational Expressions, Equations, and Functions
Vertical Asymptotes:
346
University of Houston Department of Mathematics
SECTION 5.6 Rational Functions
Finding Vertical Asymptotes
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
347
CHAPTER 5 Rational Expressions, Equations, and Functions
Horizontal Asymptotes:
348
University of Houston Department of Mathematics
SECTION 5.6 Rational Functions
Additional Example 1:
Solution:
f  x  0
x3
0
x 1
MATH 1300 Fundamentals of Mathematics
349
CHAPTER 5 Rational Expressions, Equations, and Functions
350
University of Houston Department of Mathematics
SECTION 5.6 Rational Functions
Additional Example 2:
Solution:
Additional Example 3:
MATH 1300 Fundamentals of Mathematics
351
CHAPTER 5 Rational Expressions, Equations, and Functions
Solution:
352
University of Houston Department of Mathematics
SECTION 5.6 Rational Functions
MATH 1300 Fundamentals of Mathematics
353
Exercise Set 5.6: Rational Functions
Find the indicated function values. If undefined, state
“Undefined.”
1.
If f ( x ) 
(a)
2.
3.
4.
(a)
5.
f
 13 
(c)
f
 15 
(c)
f
 54 
3x  2
, find
x7
(b) f (3)
2x  7
, find
x6
(b) f (4)
f (0)
Graph I:
Graph II:

(c)
2
, find
x2  x  6
(c)
f (2) (b) f (0)
x 1
If f ( x)  2
, find
x  2x 1
(a) f (4) (b) f (0)
(c)
y


 
f  34
y

x


x







If f ( x) 
(a)
6.
(b) f (5)
f (0)
If f ( x ) 
(c)
5
, find
x5
f (0)
If f ( x ) 
(a)
(b) f (1)
f (0)
If f ( x) 
(a)
x
, find
x3
The graph of each of the following functions has a
horizontal asymptote at y  1 . (You will learn how to
find horizontal asymptotes in a later mathematics
course.) For each function,
(a) Find the domain of the function and express it
as an inequality.
(b) Write the equation of the vertical
asymptote(s) of the function.
(c) Find the x- and y-intercept(s) of the function,
if they exist. If an intercept does not exist,
state “None.”
(d) Find f (1) and f (1) .
(e) Based on the features from (a)-(d), match the
function with its corresponding graph, using
the choices (Graphs I-IV) below.


f (5)
Graph III:
Graph IV:

f (1)

y
y


x
x
7.
x
, find
x  121
f (3) (b) f (0)
(c)
1
, find
x 2  5 x  14
(a) f (0)
(b) f (1)
(c)
If f ( x) 
(a)
8.
9.
2
f (12)
If
x3
, find
x  11x  28
(b) f (4) (c)
f (3)
If f ( x) 
(a)
(a)
f (7)
2
x5
, find
x  x  12
(b) f (2) (c)
f (0)
10. If f ( x ) 
354









11. f ( x) 
x4
x3
12. f ( x) 
x6
x2
13. f ( x) 
x6
x3
14. f ( x) 
x4
x2

f (0)
2
f (5)
University of Houston Department of Mathematics
Exercise Set 5.6: Rational Functions
The graph of each of the following functions has a
horizontal asymptote at y  0 . (You will learn how to
find horizontal asymptotes in a later mathematics
course.) For each function,
(a) Find the domain of the function and express it
as an inequality.
(b) Write the equation of the vertical
asymptote(s) of the function.
(c) Find the x- and y-intercept(s) of the function,
if they exist. If an intercept does not exist,
state “None.”
(d) Find f (1) and f (1) .
(e) Based on the features from (a)-(d), match the
function with its corresponding graph, using
the choices (Graphs I-IV) below.
Graph I:
19. If f ( x) 
10
x5
20. f ( x)  
12
x3
21. f ( x) 
x6
x2
22. f ( x ) 
x
8 x
23. f ( x) 
x3
x
24. f ( x) 
4 x
x 1
25. f ( x) 
9 x
x2  9
26. f ( x) 
x 8
x 2  16
Graph II:


y
y


x
x

For each of the following functions,
(a) Find the domain of the function and express it
as an inequality. Then write the domain of the
function in interval notation.
(b) Write the equation of the vertical
asymptote(s) of the function.
(c) Find the x- and y- intercept(s) of the function.
If an intercept does not exist, state “None."










Graph III:

Graph IV:

y
y


x




15. f ( x)  
16. f ( x )  

x





27. f ( x)  
24
x  8 x  12
2
4
x2
28. f ( x) 
2x
x  x  20
8
x
29. f ( x) 
x5
x  2x  1
x 8
x2  5x  4
2
2
17. f ( x) 
4
x
30. f ( x) 
18. f ( x) 
8
x2
31. f ( x)  
32. f ( x) 
MATH 1300 Fundamentals of Mathematics
4
x2  8x
x4
x  6x
2
355
Exercise Set 5.6: Rational Functions
356
33. f ( x) 
x2  10 x  25
5 x
34. f ( x) 
x2  7 x  18
5x
35. f ( x) 
2x
x  25
36. f ( x) 
x 1
x 2  16
37. f ( x) 
x2  5x  14
5x  7
38. f ( x) 
9 x2  1
3x  2
39. f ( x) 
25x 2  36
x2  5x  4
40. f ( x) 
x2  7 x  6
x 2  5 x  24
2
University of Houston Department of Mathematics