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Solutions Manual for Precalculus Mathematics for Calculus 7th Edition Stewart
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Solutions Manual for Precalculus Mathematics for Calculus 7th Edition Stewart
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SOLUTIONS MANUAL FORPRECALCULUS MATHEMATICS
FOR CALCULUS 7TH EDITION
STEWART
,Precalculus Mathematics for Calculus 7th Edition Stewart Solutions Manual
2 FUNCTIONS
2.1 FUNCTIONS
1. If f x x 3 1, then
(a) the value of f at x 1 is f 1 13 1 0.
(b) the value of f at x 2 is f 2 23 1 9.
(c) the net change in the value of f between x 1 and x 2 is f 2 f 1 9 0 9.
2. For a function f , the set of all possible inputs is called the domain of f , and the set of all possible outputs is called the
range of f .
x 5
3. (a) f x x 2 3x and g x have 5 in their domain because they are defined when x 5. However,
x
h x x 10 is undefined when x 5 because 5 10 5, so 5 is not in the domain of h.
55 0
(b) f 5 52 3 5 25 15 10 and g 5 0.
5 5
4. (a) Verbal: “Subtract 4, then square and add 3.”
(b) Numerical:
x f x
0 19
2 7
4 3
6 7
5. A function f is a rule that assigns to each element x in a set A exactly one element called f x in a set B. Table (i) defines
y as a function of x, but table (ii) does not, because f 1 is not uniquely defined.
6. (a) Yes, it is possible that f 1 f 2 5. [For instance, let f x 5 for all x.]
(b) No, it is not possible to have f 1 5 and f 1 6. A function assigns each value of x in its domain exactly one
value of f x.
7. Multiplying x by 3 gives 3x, then subtracting 5 gives f x 3x 5.
8. Squaring x gives x 2 , then adding two gives f x x 2 2.
9. Subtracting 1 gives x 1, then squaring gives f x x 12 .
x 1
10. Adding 1 gives x 1, taking the square root gives x 1, then dividing by 6 gives f x .
6
x 2
11. f x 2x 3: Multiply by 2, then add 3. 12. g x : Add 2, then divide by 3.
3
x2 4
13. h x 5 x 1: Add 1, then multiply by 5. 14. k x : Square, then subtract 4, then divide by 3.
3
141
Visit TestBankDeal.com to get complete for all chapters
,142 CHAPTER 2 Functions
3
15. Machine diagram for f x x 1. 16. Machine diagram for f x .
x 2
subtract 1,
subtract 2,
1 then take 0
square root 3 take reciprocal, 3
multiply by 3
subtract 1,
subtract 2,
2 then take 1
square root _1 take reciprocal, _1
multiply by 3
subtract 1,
subtract 2,
5 then take 2
square root 1 take reciprocal, _3
multiply by 3
17. f x 2 x 12 18. g x 2x 3
x f x x g x
1 2 1 12 8 3 2 3 3 3
0 2 12 2 2 2 2 3 1
1 2 1 12 0 0 2 0 3 3
2 2 2 12 2 1 2 1 3 5
3 2 3 12 8 3 2 3 3 9
19. f x x 2 6; f 3 32 6 9 6 3; f 3 32 6 9 6 3; f 0 02 6 6;
2
f 12 12 6 14 6 23 4.
20. f x x 3 2x; f 2 23 2 2 8 4 12; f 1 13 2 1 1 2 3;
3
f 0 03 2 0 0; f 12 12 2 12 18 1 98 .
1 2x 1 2 2 1 2 2 5 1 2 12 1 2a
21. f x ; f 2 1; f 2 ; f 1 0; f a ;
3 3 3 3 2 3 3
1 2 a 1 2a 1 2 a 1 3 2a
f a ; f a 1 .
3 3 3 3
x2 4 22 4 8 22 4 8 a2 4 x2 4 x2 4
22. h x ; h 2 ; h 2 ; h a ; h x ;
5 5 5 5 5 5 5 5
2
a 22 4 a 2 4a 8 x 4 x 4
h a 2 ;h x .
5 5 5 5
23. f x x 2 2x; f 0 02 2 0 0; f 3 32 2 3 9 6 15; f 3 32 2 3 9 6 3;
2
1 1 1 1 2
f a a 2 2 a a 2 2a; f x x2 2 x x 2 2x; f 2 2 .
a a a a a
1
24. h x x 1 1 1 2; h 2 2 1 5 ; h 1 1 1 1 2 5 ;
; h 1 1 1
x 2 2 2 2 1 2 2
2
1 1 1 1 1
h x 1 x 1 ;h x.
x 1 x x 1 x
x
, SECTION 2.1 Functions 143
1 1 1
1x 1 2 1 1 1 1 1 2 1
25. g x ; g 2 ; g 1 , which is undefined; g 2 ;
1x 1 2 3 3 1 1 2 1 3 3
1 2 2
1 a 1a 1 a 1 1a1 2a 2 1 x2 1 2 x2
g a ; g a 1 ;g x 1 .
1 a 1a 1 a 1 1a1 a 1 x2 1 x2
t 2 2 2 22 02 a2
26. g t ; g 2 0; g 2 , which is undefined; g 0 1; g a ;
t 2 2 2 22 02 a2
a2 2 2 a2 a12 a3
g a2 2 2 2 ; g a 1 .
a 22 a 4 a12 a1
27. k x x 2 2x 3; k 0 02 2 0 3 3; k 2 22 2 2 3 5; k 2 22 2 2 3 3;
2
k 2 2 2 2 3 1 2 2; k a 2 a 22 2 a 2 3 a 2 6a 5;
2
k x x2 2 x 3 x 2 2x 3; k x 2 x 2 2 x 2 3 x 4 2x 2 3.
28. k x 2x 3 3x 2 ; k 0 2 03 3 02 0; k 3 2 33 3 32 27; k 3 2 33 3 32 81;
3 2 3 2 a 3 3a 2
k 12 2 12 3 12 12 ; k a2 2 a2 3 a2 ; k x 2 x3 3 x2 2x 3 3x 2 ;
4
3 2
k x 3 2 x 3 3 x 3 2x 9 3x 6 .
29. f x 2 x 1; f 2 2 2 1 2 3 6; f 0 2 0 1 2 1 2;
f 12 2 12 1 2 12 1; f 2 2 2 1 2 1 2; f x 1 2 x 1 1 2 x;
f x 2 2 2 x 2 2 1 2 x 2 1 2x 2 2 (since x 2 1 0 ).
x 2 2 1 1
30. f x ; f 2 1; f 1 1; f x is not defined at x 0;
x 2 2 1 1
5 5 x 2 x2
1 1x x
f 5 1; f x 2 2 2 1 since x 2 0, x 0; f .
5 5 x x x 1x x
31. Since 2 0, we have f 2 22 4. Since 1 0, we have f 1 12 1. Since 0 0, we have
f 0 0 1 1. Since 1 0, we have f 1 1 1 2. Since 2 0, we have f 2 2 1 3.
32. Since 3 2, we have f 3 5. Since 0 2, we have f 0 5. Since 2 2, we have f 2 5. Since 3 2, we
have f 3 2 3 3 3. Since 5 2, we have f 5 2 5 3 7.
33. Since 4 1, we have f 4 42 2 4 16 8 8. Since 32 1, we have
2
f 32 32 2 32 94 3 34 . Since 1 1, we have f 1 12 2 1 1 2 1. Since
1 0 1, we have f 0 0. Since 25 1, we have f 25 1.
34. Since 5 0, we have f 5 3 5 15. Since 0 0 2, we have f 0 0 1 1. Since 0 1 2, we have
f 1 1 1 2. Since 0 2 2, we have f 2 2 1 3. Since 5 2, we have f 5 5 22 9.
35. f x 2 x 22 1 x 2 4x 4 1 x 2 4x 5; f x f 2 x 2 1 22 1 x 2 1 4 1 x 2 6.
36. f 2x 3 2x 1 6x 1; 2 f x 2 3x 1 6x 2.
2
37. f x 2 x 2 4; f x [x 4]2 x 2 8x 16.
x x f x 6x 18 3 2x 6
38. f 6 18 2x 18; 2x 6
3 3 3 3 3
39. f x 3x 2, so f 1 3 1 2 1 and f 5 3 5 2 13. Thus, the net change is f 5 f 1 13 1 12.
40. f x 4 5x, so f 3 4 5 3 11 and f 5 4 5 5 21. Thus, the net change is
f 5 f 3 21 11 10.
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