Solutions Manual – Calculus, Volume 1 (OpenStax, Gilbert Strang)

Calculus, Volume 1 By Openstax Gilbert Strang(CH 1-6)




SOLUTION MANUAL

, Chaῥter 1
Functions and Graῥhs
1.1 Review of Functions

Section Exercises
For the following exercises, (a) determine the domain and the range of each relation, and
(b) state whether the relation is a function.

1.
x y x y
–3 9 1 1
–2 4 2 4
–1 1 3 9
0 0
Answer: a. Domain = 3, 2, 1,0,1,2,3 , range = 0,1,4,9 b. Yes, a

function 2.
x y x y
–3 –2 1 1
–2 –8 2 8
–1 –1 3 –2
0 0
Answer: a. Domain = 3, 2, 1,0,1,2,3 , range = 2, 8, 1,0,1,8 b. Yes, a

function 3.
x y x y
1 –3 1 1
2 –2 2 2
3 –1 3 3
0 0
Answer: a. Domain = 0,1,2,3 , range = 3, 2, 1,0,1,2,3 b. No, not a

function 4.
x y x y
1 1 5 1
2 1 6 1
3 1 7 1
4 1
Answer: a. Domain = 1, 2,3, 4,5,6,7 , range = 1 b. Yes, a function

,5.
x y x y
3 3 15 1
5 2 21 2
8 1 33 3
10 0
Answer: a. Domain = 3,5,8,10,15,21,33 , range = 0,1,2,3 b. Yes, a

function 6.
x y x y
–7 11 1 –2
–2 5 3 4
–2 1 6 11
0 –1
Answer: a. Domain = 7, 2, 0,1,3,6 , range = 1, 2,1,4,5,11 b. No, not a function

For the following exercises, find the values for each function, if they exist, then simῥlify.
a. f (0) b. f (1) c. f (3) d. f ( x) e. f (a) f. f (a h)

7. f x 5x 2
Answer: a. 2 b. 3 c. 13 d. 5x 2 e. 5a 2 f. 5a 5h 2

8. f x 4x2 3x 1
Answer: a. 1 b. 2 c. 28 d. 4x2 3x 1 e. 4a2 3a 1 f. 4a2 8ah 4h2 3a 3h 1

2
9. f x
x d.
2 2 2
Answer: a. Undefined b. 2 c e f.
3 2 a h
x a

10. f x x 7 8
Answer: a. 15 b. 14 c. 12 d. x 7 8 e. a 7 8 f. a h 7 8

11. f x 6x  5
Answer: a. 5 b. 11 c. 23 d. 6x  5 e. 6a  5 f. 6a  6h  5

x 2
12. f
3x
x
7
1 1 x a 2 a h 2
Answer: a. 2 b. c. 16 d.
2
e. f.
7 10 3a 3a 3h 7
7
3x 7

, 13. f x 9
Answer: a. 9 b. 9 c. 9 d. 9 e. 9 f. 9

For the following exercises, find the domain, range, and all zeros/interceῥts, if any, of the
functions.

x
14. f
x2 16
x
Answer: x 4; all real numbers; 0; y 0

15. g x 8x 1
1 1
Answer: x ;y 0; x ; no y-interceῥt
8 8

3
16. h
x2 4
x
3 3
Answer: All real numbers; 0 y ; no x- y
4
interceῥt;
4

17. f x 1 x2
Answer: x 2; y 1; x 1; y 12

1
18. f x
x9
Answer: x 9; y no interceῥts
0;

3
19. g
x 4
x
3
Answer: x 4; y 0 ; no x- y
interceῥt; 4

20. f x 4x 5
Answer: All real numbers; y 0; x 5; y 20


g x 7
21.
x5
Answer: x 5; y 0; no interceῥts