Active Maths 1 – Strands 1–5 – Ch 30 Solutions 1
Chapter 30 Exercise 30.1 Q. 1. Functions are: (i), (ii), (iv)
Reason: Each input is assigned to exactly one output.
Not functions: (iii), (v), (vi).
Reason: In each diagram there is an input mapped to two outputs. Also, in (v) there is an input mapped to no output.
Q. 2. B and C are functions as in each, each input is mapped to exactly one output.
Q. 3. Outputs: 8, 9, 10, 12.
Q. 4. Outputs: 3, 5, 9, –5.
Q. 5. Domain Codomain Range Function
(i)
(ii)
(iii)
(iv)
{0, 1, 2, 3}
{2, 5, 7, 9}
{–3, 3, 4}
{1, 2, 4, 6}
{0, 3, 4, 6, 9}
{4, 6, 7, 8, 11}
{5, 8, 13, 20}
{5, 7, 8, 11, 15, 19}
{0, 3, 6, 9}
{4, 7, 11}
{13, 20}
{5, 7, 8}
Yes
Yes
Yes
No
Q. 6. (i) Domain: {1, 2, 3, 5}
Range: {10, 11, 12, 14}
(ii)
1
2
3
5 14
11
12
10
(iii) Yes. Each input is mapped to exactly one output.
Q. 7. A: Change (6,5) to (5,5), for example.
B: Change (–1,5) to (0,5), for example.
C: Change (2,12) to (3,12), for example.
D: Change (3,1) to (5,1), for example.
Q. 8. Range = {5, –4, 10, 25}
Q. 9. {(1,0), (–3,8), (–5,24), (5,24)}
Q. 10. (i) ‘Add 6’
(ii) ‘Subtract 39’
(iii) ‘Square the input’
(iv) ‘Double the input and subtract 4’
Q. 11. (i) Domain: {0, 1, 3, –3}
Range: {0, 1, –27, –1}
(ii)
0
1
3
–3 –27
1
–1
0
(iii) No, because 1 is mapped to two outputs.
Q. 12.
Input Rule: Add 5 Output0 0 + 5 = 5 52 2 + 5 = 7 74 4 + 5 = 9 96 6 + 5 = 11 11
Range: {5, 7, 9, 11}
Couples: {(0,5), (2,7), (4,9), (6,11)}
Q. 13.
Input Rule: Subtract 5 Output10 10 – 5 = 5 55 5 – 5 = 0 0
50 50 – 5 = 45 45–5 –5 – 5 = –10 –10
Range: {5, 0, 45, –10}
Couples: {(10,5), (5,0), (50,45), (–5,–10)}
Q
Q
Q
Q
Q
2 Active Maths 1 – Strands 1–5 – Ch 30 Solutions
Q. 14. Input Rule: Double the input and add 3
Output
–3 2(–3) + 3 = –6 + 3= –3 –3
1 2(1) + 3 = 2 + 3 = 5 54 2(4) + 3 = 8 + 3 = 11 117 2(7) + 3 = 14 + 3 = 17 17
Range: {–3, 5, 11, 17}
Couples: {(–3,–3), (–1,5), (4,11), (7,17)}
Q. 15. Input Rule: Triple the input and subtract 3
Output
0 3(0) – 3 = 0 – 3 = –3 –3
4 3(4) – 3 = 12 – 3 = 9 9
–6 3(–6) – 3 = –18 – 3 = –21 –21
6 3(6) – 3 = 18 – 3 = 15 15
Range: {–3, 9, –21, 15}
Couples: {(0,–3), (4,9), (–6,–21), (6,15)}
Q. 16. Input Rule: Square the input and add 1
Output
3 (3)2 + 1 = 9 + 1 = 10 10
1 (1)2 + 1 = 1 + 1 = 2 2
–4 (–4)2 + 1 = 16 + 1 = 17 17
–3 (–3)2 + 1 = 9 + 1 = 10 10
Range: {10, 2, 17}
Couples: {(3,10), (1,2), (–4,17), (–3,10)}
Q. 17.
5 –9
–11
7
3
3 –5
6
–3
–1
Q. 18.
8 125
5
47
197
0 –3
–2
5
–10
Q. 19. (i) Domain = {2, 4, –4, 6}
Range = {1, 8, 18}
(ii)
2
4
–4
6
8
18
1
(iii) Yes, as each input is mapped to exactly one output.
Q. 20.
Input Rule Output0 (0)2 + 4 4
–5 (–5)2 + 4 29–12 (–12)2 + 4 148
2 (2)2 + 4 8
Ordered pairs: {(0,4), (–5,–29), (–12,148), (2,8)}
Q. 21.
Input Rule Output0 (0)2 – 6 –67 (7)2 – 6 43
–5 (–5)2 – 6 19–14 (–14)2 – 6 190
Ordered pairs: {(0,–6), (7,43), (–5,19), (–14,190)}
Q. 22. Input Rule Output0 (0)3 – 10 –101 (1)3 – 10 –92 (2)3 – 10 –2
–2 (–2)3 –10 –18
Ordered pairs: {(0,–10), (1,–9), (2,–2), (–2,–18)}
Q. 23.
Input Rule Output0 2(0)2 + 5 51 2(1)2 + 5 74 2(4)2 + 5 378 2(8)2 + 5 133
Ordered pairs: {(0,5), (1,7), (4,37), (8,133)}
Q
Q
Q
Q
Q
Q
3Active Maths 1 – Strands 1–5 – Ch 30 Solutions
Q. 24. {4, 15, –10, –30}
Q. 25. 4 + 2 = 6 12 + 2 = 14 –4 + 2 = –2 0 + 2 = 2
6 ÷ 2 = 3 14 ÷ 2 = 7 –2 ÷ 2 = –1 2 ÷ 2 = 1
∴ {3, 7, –1, 1}
Q.26. (a) (i) 1.8(0) + 32
= 32°F
(ii) 1.8(16) + 32
= 60.8°F
(iii) 1.8(–2) + 32
= 28.4°F
(b) (i) 32 – 32 = 0
0 ÷ 1.8 = 0
Answer = 0°C
(ii) 104 – 32 = 72
72 ÷ 1.8 = 40
Answer = 40°C
(iii) 23 – 32 = –9
–9 ÷ 1.8 = –5
Answer = –5°C
Q. 27. (a) (i) 1 m2 (ii) 9 m2 (iii) 25 m2
(b) √____
225 = 15 hrs
Exercise 30.2
Q. 1. (i) 0 (ii) 6 (iii) 8
Q. 2. (i) – 4 (ii) 16 (iii) 48
Q. 3. (i) 1 (ii) 5 (iii) –2 (iv) – 6
Q. 4. (i) – 8 (ii) 12 (iii) –24 (iv) – 8
Q. 5. (i) 0 (ii) 1 (iii) –1 (iv) 4
Q. 6. x → x + 5
Q. 7. x → x – 5
Q. 8. x → 2x + 3
Q. 9. x → 3x – 3
Q. 10. x → 4x – 1
Q. 11. ‘Subtract 3 from the input.’
Q. 12. ‘Double the input and then add 1.’
Q. 13. ‘Multiply the input by –4 and then add 3.’
Q. 14. ‘4 times the input and then subtract 11.’
Q. 15. ‘Square the input and then add 3.’
Q. 16. ‘Twice the input squared and then subtract 4.’
Q. 17. ‘5 times the input squared and then add the input.’
Q. 18. ‘Square the input, then subtract 3 times the input and add 2.’
Q. 19. (i) g(–2) = –9 (iii) g(10) = 51
(ii) g(– 4) = –19 (iv) g(6) = 31
–2
–4
10
6 31
–19
51
–9
Q. 20. (i) g(–2) = 0 (iii) g(10) = 36
(ii) g(–4) = –6 (iv) g(6) = 24
–2
–4
10
6 24
–6
36
0
Q. 21. (i) g(–2) = 6 (iii) g(10) = – 6
(ii) g(–4) = 8 (iv) g(6) = –2
–2
–4
10
6 –2
8
–6
6
Q
Q
Q
Q
Q
E i 30 2 Q 17 ‘5 ti th i t d d
4 Active Maths 1 – Strands 1–5 – Ch 30 Solutions
Q. 22. (i) g(–2) = –22 (iii) g(10) = 38
(ii) g(–4) = –32 (iv) g(6) = 18
–2
–4
10
6 18
–32
38
–22
Q. 23. (i) g(–2) = 8 (iii) g(10) = –16
(ii) g(–4) = 12 (iv) g(6) = – 8
–2
–4
10
6 –8
12
–16
8
Q. 24. x → x2
Q. 25. x → x2 – 6
Q. 26. x → x3 – 10
Q. 27. x → 2x2
Q. 28. x → x2 + 2x
Q. 29. (i) h(–3) = 10
(ii) h(0) = 1
(iii) h(1) = 2
(iv) h(4) + 2h(1) = 17 + 4 = 21
Q. 30. (i) h(–3) =9
(ii) h(0) = 3
(iii) h(1) = 5
(iv) h(4) + 2h(1) = 23 + 10 = 33
Q. 31. (i) h(–3) =35
(ii) h(0) = 5
(iii) h(1) = 3
(iv) h(4) + 2h(1) = 21 + 6 = 27
Q. 32. (i) h(–3) = 33
(ii) h(0) = –9
(iii) h(1) = –7
(iv) h(4) + 2h(1) = 47 – 14 = 33
Q. 33. (i) h(–3) = –18
(ii) h(0) = 0
(iii) h(1) = 2
(iv) h(4) + 2h(1) = – 4 + 4 = 0
Q. 34. (i) f(0) = 4(0) – 8 = 0 – 8 = –8
(ii) f(–3) = 4(–3) – 8 = –12 – 8 = –20
(iii) f(0) + f(–2) = –8 + 4(–2) – 8 = –8 – 8 – 8 = –24
(iv) 2[f(1)]2 = 2[4(1) – 8]2
= 2[4 – 8]2
= 2[–4]2
= 2(16) = 32
Q. 35. (i) f(4) = (4)2 – 2 = 16 – 2 = 14
(ii) f(2) = (2)2 – 2 = 4 – 2 = 2
(iii) f(0) = (0)2 – 2 f(–1) = (–1)2 – 2 = 0 – 2 = 1 – 2 = –2 = –1 –2 < – 1 ∴ f(0) < f(–1) Answer: Yes
(iv) f(1) = (1)2 – 2 f(–3) = (–3)2 – 2 = 1 – 2 = 9 – 2 = – 1 = 7 ∴ 3f(1) – 2f(–3) = 3(–1) – 2(7) = –3 – 14 Answer = –17
Q. 36. (a) (i) f(0) = –1
(ii) f(3) = 3(3)2 + 3 – 1
= 3(9) + 2
= 27 + 2 = 29
(iii) f(0) = –1
[f(0)]2 = 1
5[f(0)]2= 5
f(– 4) = 3(– 4)2 + (– 4) –1
= 3(16) – 4 – 1
= 48 – 5 = 43
∴ 5[f(0)]2 – f(–4) = 5 – 43
Answer = –38
Q
Q
Q
5Active Maths 1 – Strands 1–5 – Ch 30 Solutions
(b) f(2) = 3(2)2 + (2) – 1 f(–2) = 3(–2)2 + (–2) –1 = 3(4) + 2 – 1 = 3(4) – 2 – 1 = 12 + 1 = 12 – 3 = 13 = 9
∴ f(2) – f(–2) = 13 – 9 = 4 ≠ 0 Answer: f(2) – f(–2) ≠ 0
Q. 37. (i) 0 (ii) 10 (iii) 8 (iv) 5 + 2(2) = 9
Q. 38. y ∈ {4, 12, –16, 0}
Q. 39. g(0) = 3, g(–1) = 5, g(5) = –7, g(–6) = 15, g(7) = –11.
∴ Range: {3, 5, –7, 15, –11}
∴ Couples: {(0,3), (–1,5), (5,–7), (– 6,15), (7,–11)}
Q. 40. Range: {–2, 3, 10, 19} Couples: {(1,–2), (2,3), (3,10), (4,19)}
Q. 41. Range: {–25, –9, 0, 11} Couples: {(0,–25), (4,–9), (5,0), (6,11}
Q. 42. Range: {12, 21, –3, 5} Couples: {(–3,12), (–4,21), (0,–3), (4,5)}
Q. 43. a = 10, b = 14, c = –6, d = 8
Q. 44. 3(0) – 4 = d 3(2) – 4 = e 3(f) – 4 = –10 3(g) – 4 = 11
0 – 4 = d 6 – 4 = e 3f = –6 3g = 15
d = –4 e = 2 f = –2 3g = 15
Q. 45. r3 – 1 = –2 s3 – 1 = 7 13 – 1 = q 43 – 1 = p
r3 = –1 s3 = 8 0 = q 64 – 1 = p
r = –1 s = 2 q = 0 p = 63
Q. 46. x2 – 4 = 21 (0)2 – 4 = c x2 – 4 = 0
x2 = 25 0 – 4 = c x2 = 4
x = ± 5 c = –4 x = ± 2
a = 5 d = 2
b = –5 e = –2
Q. 47. Domain = {0, 1, 5, 16}
Q. 48. 3 – 5x = –2 3 – 5x = –47 3 – 5x = 13 3 – 5 x = 3
–5x = –5 –5x = –50 –5x = 10 –5x = 10
5x = 5 5x = 50 5x = –10 5x = 0
x = 1 x = 10 x = –2 x = 0
Domian = {1, 10, –2, 0}
6 Active Maths 1 – Strands 1–5 – Ch 30 Solutions
Q. 49. f(–1) = (–1)2 + 1
= 1 + 1 = 2
f(0) = (0)2 + 1
= 0 + 1 = 1
f(1) = (1)2 + 1
= 1 + 1
= 2
(i) Range = {1, 2}
(ii) Codomain = {0, 1, 2}
Q. 50. (i) 6x – 12 = 18
6x = 30
x = 5
(ii) 6x – 12 = 0
6x = 12
x = 2
(iii) 6(a) – 12 = –12
6a = 0
a = 0
Q. 51. (a) (i) 8(3) + 30
= 24 + 30
= ¤54
(ii) 8(6) + 30
= 48 + 30
= ¤78
(ii) 8(11) + 30
= 88 + 30 = ¤118
(b) 8(0) + 30
= 0 + 30
= ¤30
(c) 8x + 30 = 350
8x = 320
x = 40 weeks
(d) ¤8
Q. 52. (a) (i) 2(100) – 100 = ¤100
(ii) 2(50) – 100 = ¤0
(iii) 2(75) – 100 = ¤50
(iv) 2(125) – 100 = ¤150
(b) Phone B
Q. 53. (a) (i) (0)2 – 3(0) + 2 = 2˚C
(ii) (1)2 – 3(1) + 2 = 0˚C
(iii) (2)2 – 3(2) + 2 = 0˚C
(iv) (3)2 – 3(3) + 2 = 2˚C
(v) (4)2 – 3(4) + 2 = 6˚C
(b) x = –2
(–2)2 – 3(–2) + 2
4 + 6 + 2
12˚C
Revision Exercises
Q. 1. f(3) = 5(3) + 2
= 15 + 2
= 17
Q. 2. g(3) = 7(2) – 1
= 14 – 1
= 13
Q. 3. f(7) = 10(7) + 3
= 73
Q. 4. g(5) = 6(5) – 10
= 20
Q. 5. 2(4) + 1 = 9
2 ( 1 1 __ 2 ) + 1 = 4
2x + 1 = 21
2x = 20
x = 10
Q. 6. 4(5) – 3 = 20 – 3
= 17
4(–2) – 3 = –8 – 3
= –11
4x – 3 = 13
4x = 16
x = 4
∴ a = 17
b = –11
c = 4
Q
R
4
11–
2
10 21
9
4
f
7Active Maths 1 – Strands 1–5 – Ch 30 Solutions
Q. 7. p = 5, q = 0, r = –2, s = 8
Q. 8. 5(1) – 2 = 3
5(2) – 2 = 8
5(3) – 2 = 13
Range = {3, 8, 13}
Q. 9. (i) f(6) = 10 – 6
= 4
(ii) Range = {4, 3, 2, 1, 0, –1}
Q. 10. Range = {5, 2, 1}
Q. 11. Range = {14, 17, 20}
Q. 12. Range = {–10, –3, 4, 11}
Q. 13. Range = {29, 4}
Q. 14. (i) f(3) = 18
(ii) f(5) = 50
(iii) f(10) = 200
Q. 15. Range = {7, 22, 43, 70}
Q. 16. (i) 7 (ii) 37 (iii) 87
Q. 17.
(i) 27 (iii) 5p – 8 = 17
5p = 25
p = 5
(ii) –43 (iv) 5q – 8 = q
4q = 8
q = 2
Q. 18. (i) g(1) = 5, g(2) = 7, g(3) = 9
(ii) 9 = 5 + 7
9 = 12
False
∴ g(3) ≠ g(1) + g(2)
(iii) 2x + 3 = 53
2x = 50
x = 25
Q. 19. 3(2)3 – 5 = 19
3(–3)3 – 5 = –86
3x3 – 5 = 2995
3x3 = 3000
x3 = 1000
x = 10
f(2) = 19, f(–3) = –86, f(10) = 2995
2
10
–3
2995
19
–86
f
Q. 20. 5(2)2 + 1 = 21
5(–2)2 + 1 = 21
5x2 + 1 = 46
5x2 = 45
x2 = 9
x = ±3
f(–2) = f(2) = 21, f(–3) = f(3) = 46
2
–2
3
–3 46
21
f
Q. 21. (i) 2(0) + 1
2(1) + 1
2(2) + 1
Range = {1, 3, 5}
(ii) Codomain = {0, 1, 2, 3, 4, 5}
Q. 22. (i) (–1)2 + 2
(0)2 + 2
(1)2 + 2
Range = {3, 2}
(ii) Codomain = {0, 1, 2, 3}
Q. 23. f(5) = (5)2 + 2(5) + 3
= 25 + 10 + 3
= 38
Q
Q
Q
Q
Q
8 Active Maths 1 – Strands 1–5 – Ch 30 Solutions
Q. 24. (i) 100 – 2(5)2
= 100 – 50
= 50
(ii) 100 – 2(7)2
= 100 – 98
= 2
(iii) 100 – 2(2)2
= 100 – 8
= 92
Q. 25. (3)2 – (3) – 6
= 9 – 3 – 6
= 0
∴ f(3) = 0
(–2)2 – (–2) – 6
= 4 + 2 – 6
= 0
∴ f(–2) = 0
Q. 26. (i) f(5) = (9)(2)
= 18
(ii) f(3) = (5)(0)
= 0
(iii) f ( 1 __ 2 ) = (0) ( –2 1 __ 2 ) = 0
Q. 27. (i) f(4) = (2)(5)
= 10
(ii) f(6) = (0)(7)
= 0
(iii) f(–1) = (7) (0)
= 0
Q. 28. (i) 3 + 1 ______ 2 = 2 hrs (ii) 2 + 1 ______ 2 = 1 1 __ 2 hrs
Q. 29. 2(6) + k = 22
12 + k = 22
k = 10
Q. 30. (i) a(5) + 7 = 22
5a = 15
a = 3
(ii) 3x + 7 ≤ 16
3x ≤ 9
x ≤ 3, x ∈ N
–1 1 2 3 4 50
Q
Q
Q
Q
