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to let us know that this function is the inverse of f. World View Note: The notation used for functions was ο¬rst introduced by the great Swiss mathematician, Leonhard Euler in the 18th century. 1(x) = x For example, if f (x) = x + 5, we could deduce that the inverse function would be 5. If we had an input of 3, we cou... |
and g(x) = x + 2 4x 3 β inverses? Calculate composition f (g(x)) Replace g(x) with x + 2 4x 3 β f Substitute x + 2 4x 3 β x + 2 4x 3 β for variable in f x + 2 4x 3 β x + 2 4x 3 β 3 4 Distribute 3 and 4 into numerators 2 β + 1 402 2 3x + 6 3 β 4x + 8 3 4x β 4x + 1 β (3x + 6)(3 4x 3 (4x + 8)(3 4x 3 β β β β 4x) 4x) 2(3 β... |
inverse of g(x) = 2x 3 β 4x + 2 Replace g(x) with y y = 2x 3 β 4x + 2 x = 2y 3 β 4y + 2 Switch x and y Multiply by (4y + 2) x(4y + 2) = 2y 4xy + 2x = 2y 4xy + 3 2x + 3 = 2y β 4xy + 3 4xy β β 3 Distribute 3 Move all y β²s to one side, rest to other side Subtract 4xy and add 3 to both sides Factor out y β β 2x + 3 = y(2 ... |
2 x + 2 6) f (x) = x 5 β 10 h(x) = 10x + 5 5 8) f (x) = x + 1 2 g(x) = 2x5 q 1 β 10) g(x) = 8 + 9x f (x) = 5x 9 2 β 2 11) f (x) = (x 2)5 + 3 12) g(x) = x + 1 3β + 2 β x β 5 (x β 1 13) g(x) = 4 x + 2 15) f (x) = β 2x β x + 2 2 17) f (x) = 10 1)3 β 3)3 19) g(x) = β 21) f (x) = (x 23) g(x) = x x β 25) f (x) = x 1 β x + 1... |
will look at exponential functions and then we will consider logarithmic functions in another lesson. Exponential functions are functions where the variable is in the exponent such as f (x) = ax. (It is important not to confuse exponential functions with polynomial functions where the variable is in the base such as f... |
34)4x+1 Multiply exponents 3(3x + 5) and 4(4x + 1) 39x+15 = 316x+4 Same base, set exponents equal 9x + 15 = 16x + 4 Move variables to one side 9x Subtract 9x from both sides Subtract 4 from both sides β β 9x 15 = 7x + 4 4 4 β β 7 11 = 7x Divide both sides by 7 7 11 7 = x Our Solution Another useful exponent property is... |
(2β 6x 20 Β· 28x β Β· 2 = 25 2β Β· x+3 x+3 1 2 1 Β· (2β Write with a common base of 2 Multiply exponents, distributing as needed β 22x x 2β β Β· x+2 22 = 2β 3 Add exponents, combining like terms Same base, set exponents equal β 22 = 2x + x β β + x x + 2 Move variables to one side Add x to both sides 22 = 2 Add 22 to both s... |
) 4β 3v = 64 18) 64x+2 = 16 20) 162k = 1 64 22) 243p = 27β 3p 24) 42n = 42 β 3n 26) 6252x = 25 28) 2162n = 36 30) ( 1 4)3v 2 = 641 β β v 32) 216 6β2a = 63a 34) 322p β 2 8p = ( 1 2 )2p Β· 36) 32m 38) 32 β Β· x 33m = 1 33m = 1 Β· 40) 43r 4β 3r = 1 64 Β· 409 10.5 Functions - Logarithmic Functions Objective: Convert between lo... |
Identify base, 7, answer, b, and exponent, 2 log7b = 2 Our Solution 4 = 2 3 log 2 3 16 81 16 81 Identify base, 2 3, answer, 16 81, and exponent 4 = 4 Our Solution Example 536. Write each logarithmic equation in exponential form log416 = 2 Identify base, 4, answer, 16, and exponent, 2 42 = 16 Our Solution 410 log3x = 7... |
(3x + 5) = 4 Change to exponential form 24 = 3x + 5 Evaluate exponent 16 = 3x + 5 Solve 5 5 Subtract 5 from both sides β β 3 11 = 3x Divide both sides by 3 3 11 3 = x Our Solution logx8 = 3 Change to exponential form x3 = 8 Cube root of both sides x = 2 Our Solution There is one base on a logarithm that gets used more ... |
log3 243 24) log8 k = 3 26) log n = 3 28) log4 p = 4 β 4) = β 8r = 1 30) log11 (x 32) log2 β 34) log7 β 36) log11 (10v + 1) = 3n = 4 38) log9 (7 β 40) log8 (3k 6x) = β 1) = 1 β 1 1 β 2 413 10.6 Functions - Compound Interest Objective: Calculate ο¬nal account balances using the formulas for compound and continuous inter... |
will your balance be at the end of those ο¬ve years? P = 25000, r = 0.065, n = 4, t = 5 5 4 A = 25000 1 + Β· 0.065 4 A = 25000(1.01625)4 Identify each variable Plug each value into formula, evaluate parenthesis 5 Multiply exponents Β· 414 A = 25000(1.01625)20 Evaluate exponent A = 25000(1.38041977 ) Multiply A = 34510.49... |
A = 4000 1 + Β· 0.03 365 Identify each variable Plug each value into formula, evaluate parenthesis A = 4000(1.00008219 )365 7 Multiply exponent A = 4000(1.00008219 )2555 Evaluate exponent Β· A = 4000(1.23366741.) Multiply A = 4934.67 S4934.67 Our Solution While this diο¬erence is not very large, it is a bit higher. The t... |
say he used e because his name begins with E. Others, say it is because exponent starts with e. Others say it is because Eulerβs work already had the letter a in use, so e would be the next value. Whatever the reason, ever since he used it in 1731, e became the natural base. Example 548. If S4000 is invested in an acc... |
for 12 years. 418 j. All of the above compounded continuously. 2) What principal will amount to S2000 if invested at 4% interest compounded semiannually for 5 years? 3) What principal will amount to S3500 if invested at 4% interest compounded quarterly for 5 years? 4) What principal will amount to S3000 if invested at... |
ent, but we will not need to use them here). To the right is a picture of a right triangle. Based on which angle we are interested in on a given problem we will name the three sides in relationship to that angle. In the picture, angle A is the angle we will use to name the other sides. The longest side, the side opposi... |
6 10 = 3 5 5 adjacent = 6 tanΞΈ = opposite cosΞΈ = adjacent hypotenuse = 8 10 = 4 8 = 3 Now we will ο¬nd the three ratios of Ξ±. from Ξ±, the The hypotenuse is 10, opposite side is 8 and the adjacent side is 6. So we ο¬ll in the following: 4 8 ΞΈ 6 Ξ± 10 6 Opposite of ΞΈ Adjacent of Ξ± Ξ± Adjacent of ΞΈ Opposite of Ξ± 8 ΞΈ 10 Hypot... |
is the cosine. So we will take the cosine of our angle. Cosine is adjacent over hypotenuse Evaluate cos70β¦, put over 1 so we have a proportion x 70β¦ 9 cos70β¦ = 0.342 1 = x 9 x 9 3.08 = 1x 3.08 = x Our Solution. Find the cross product. 422 10.7 Practice - Trigonometric Functions Find the value of each. Round your answe... |
A B C 13.1 40β¦ x B 38) A 40) C 1.4 65β¦ B x C 18.1 A 35.5β¦ x B 427 10.8 Functions - Inverse Trigonometric Functions Objective: Solve for missing angles of a right triangle using inverse trigonometry. We used a special function, one of the trig functions, to take an angle of a triangle and ο¬nd the side length. Here we w... |
οΏ½nd angle sinβ 1(0.706) Evaluate fraction, take sine inverse using table or calculator 45β¦ Our Solution Example 556. Find the indicated angle 5 3 Ξ± From the angle Ξ±, the given sides are the opposite (5) and the adjacent (3) The trig function that uses opposite and adjacent is the tangent As we are looking for an angle ... |
nd the leg. Tangent is opposite over adjacent 5 tan35β¦ = x 5 0.700 x 1 0.700x = 5 0.700 0.700 Divide both sides by 0.700 Find cross product = Evaluate tangent, put it over one so we have a proportion x = 7.1 The missing leg. a2 + b2 = c2 We can now use pythagorean thorem to ο¬nd hypotenuse, c 52 + 7.12 = c2 Evaluate exp... |
degree. 1) sin Z = 0.4848 3) sin Y = 0.6561 2) sin Y = 0.6293 4) cos Y = 0.6157 Find the measure of the indicated angle to the nearest degree. 5) 7) 9) 35? 32 30? 31 3? 6 6) 8) 39? 46 24? 8 10) 16 23? 432 Find the measure of each angle indicated. Round to the nearest tenth. 11) B 8 A ΞΈ 8 C 13) 15) A ΞΈ 11 C B 11 B 7 ΞΈ ... |
7 48) 3 49) 2 50) 5 51) 2 52) 9 53) 7 54) 10 β 55) 4 56) 10 8 57) β 58) 6 59) 60) 6 9 β β 22) 0 23) 11 24) 9 25) 26) β β 27) β 28) 4 29) 0 30) 31) 32) β β β 33) β 34) 14 35) 8 36) 6 3 4 3 8 4 35 80 56 6 36 10 37) 38) β β 39) β 40) 63 41) β 42) 4 Answers - Fractions 4) 8 3 5) 3 2 6) 5 4 438 7) 5 4 8) 4 3 9) 3 2 10) 8 3... |
7) 25 8) 46 9) 7 10) 8 11) 5 12) 10 13) 1 14) 6 15) 1 16) 2 17) 36 18) 54 Answers - Order of Operation 6 10 9 22 10) 11) β β 12) β 13) 20 14) β 15) 2 16) 28 17) 18) 40 15 β β Answers - Properties of Algebra 19) 7 20) 38 21) r + 1 4x 2 β 22) β 23) 2n 24) 11b + 7 25) 15v 26) 7x 27) 9x β 7a 1 β 28) β 29) k + 5 30) 31) 32... |
78) β β v2 + 2v + 2 7b2 + 3b 8 β 79) 4k2 + 12 β 80) a2 + 3a 81) 3x2 15 β n2 + 6 β 5 82) β β 1.1 1) 7 2) 11 5 3) β 4) 4 5) 10 6) 6 7) 19 β 6 8) β 9) 18 10) 6 11) 12) 20 7 β β 108 13) β 14) 5 1.2 1) 4 β Answers - Chapter 1 Answers to One-Step Equations 8 15) β 16) 4 17) 17 18) 4 19) 20 20) β 21) 3 22) 16 208 23) 13 β 9 ... |
numbers 47) No Solution 48) 1 9 49) β 50) 0 1.4 1) 3 4 2) β 3) 6 5 4) 1 6 5) β 6) 25 8 7) 8) β β 9) β 10) 3 2 1.5 4 3 19 6 7 9 1 3 2 1. b = c a 2. h = gi 3. x = gb f 4. y = pq 3 5. x = a 3b 6. y = cb dm 7. m = E c2 8. D = ds S 9. Ο = 3V 4r3 10. m = 2E v2 Answers to Solving with Fractions 3 2 4 3 5 3 11) 0 12) 4 3 13) ... |
β x β 5 45. y = 7 3x β 2 46. a = 7b + 4 5 47. b = 5a 4 β 7 48. x = 8 + 5y 4 49. y = 4x 8 β 5 44. x = 7 2y β 3 50. f = 9c + 160 5 Answers to Absolute Value Equations 1.6 1) 8, 2) 7, 3) 1, 4) 2, 5) 6, 6) 38 9 7) β 8) β 9) 3 29 4 6, β 2, β 3, 9 39 7 β, β 6 29 3 β 1, 3 β 9, 15 10) 16 5 11) 7, 12) 13) 1.7 β β 5 3 β 2, 0 14... |
6) 62 7) 16 8) 17 4 9) 35, 36, 37 10) β 11) 43, 14, β 12) 52, 54 42, 13, 41 12 β β β β 13) 61, 63, 65 14) 83, 85, 87 15) 9, 11, 13 16) 56, 56, 68 1.9 1) 6, 16 2) 10, 40 3) 18, 38 24) 3.5 hours 25) 4.29 dollars 26) 450 m 27) 40 kg 28) 5.7 hr 29) 40 lb 30) 100 N 31) 27 min 33) r = 36 34) 8.2 mph 35) 2.5 m 36) V = 100.5 ... |
) 16, 32 31) 10, 28 32) 12,20 33) 141, 67 34) 16, 40 35) 84, 52 36) 14, 42 37) 10 38) 10, 6 39) 38, 42 40) 5 Answers - Distance, Rate, and Time Problems 14) 10 15) 2 16) 3 17) 48 18) 600 19) 6 20) 120 21) 36 22) 2 23) 570 24) 24, 18 25) 300 26) 8, 16 27) 56 28) 95, 120 29) 180 30) 105, 130 31) 2:15 PM 32) 200 33) 1 3 3... |
) y = β 11) y = x 12) y = 13) y = 14 4x 3 β 3 4 x + 2 1 10 x 15) y = β 3 10 β 16) y = 1 10x β 2x 1 β x + 70 11 17) y = β 18) y = 6 11 19 20) y = β 21) x = β 22) y = 1 7 8 x + 6 37 10 x 23) y = β 24) y = 5 x 2 1 β 25) y = 4x x + 1 2 3 4x + 3 26) y = β 27) y = β 28) x = 4 2.4 Answers - Slope-Intercept 1 2 x + 1 37) 29) y... |
11 4 3 2 1 10 43) y = 44) y = β β 45) y = β 46) y = 1 2 x 47) y = β 48) y = 1 3 x + 1 49) y = x + 2 β 50) y = x + 2 51) y = 4x + 3 52 22) y = 23) y = 24) y = 7 4 3 2 5 2 β β β 25) y = β 26) y = 7 x 3 27 28) y = β 3 β 3 29) x = β 30) y = 2x 31) y = β 32) y = 6 x 5 3 = 33) y β 34(x + 4) β 35) y 36) y β β 1 = 1 8(x 5 = β... |
) 20) v > 1: [1, β 21) x > 11: [11, β β ) ) β 2] 22) x 6 18: ( β 23) k > 19: (19, β β ) β 18], β 10: (, 10] 24) n 6 25) p < β β 1: ( 26) x 6 20: ( β β, β 1) β β β, 20] 27) m > 2: [2, 28) n 6 5: (, 5] 29) r > 8: (8], β 30) x 6 3: ( β 31) b > 1: (1, 32) n > 0: [0, β β ) β ) β β 26, ) ) β 13) x > 110: [110, 26: [ 14) n > ... |
3: ( β 8) 7 < x < 4: ( 8, 3) 7, 4) β 9) b < 5: ( β, 5) β β 2 6 n 6 6: [ 10) β 11) 7 6 a 6 6: [ β 12) v > 6: [6, ) β 2, 6] 7, 6] β β 13) β 14) 2: [ 2] β 6, β 9, 0] β β 15) 3 < k 6 4: (3, 4] 16) 17) β β 2 6 n 6 4: [ 2 < x < 2: ( 2, 4] 2, 2) β β 18) No solution : β 1 6 m < 4: [ 19) β 20) r > 8 or r < β 21) No solution : ... |
( 175, ) S 5 3] S h [9, 1) S (0, ) β S 1) (54, β ) β [3, ) β S [14 ) β 18) ( 19) ( 20) ( 21) ( 22) [ 23) ( 24) (, β, 2 3) β β β β β β, 0) S 1],, β β 4] 5 2] β β β β 25) [1, 3] 26) [ 1 2, 1] 27) ( 4), β β β ( 3, ) β β 28) [3, 7] 29) [1, 3 2 ] 30) [ 2, β 31) ( β β 4 3] β, 3 2) S ( 5 2 32) ( 1 S 2), β β β 33) [2, 4] S 34... |
) No Solution 3, 1) 2) 28) ( β 29) (4, β 30) (1, 4) 29) (4, 30) ( β 3) β 1, 5) 31) (0, 2) 7) 32) (0, β 33) (0, 3) 4) 2) 3) 34) (1, 35) (4, β β 36) (8, β 37) (2, 0) 38) (2, 5) 39) ( 4, 8) β 40) (2, 3) 3, 4) 1) 19) ( β 20) (2, β 21) (3, 2) 4, 22) ( 4) β β Answers - Substitution 15) (1, 16) ( 17) ( β β 5) β 1, 0) 1, 8) 18... |
) 4) 21) (0, β 22) (0, 1) 23) ( β 24) (2, 2, 0) 2) β 14) solutions β 15) (2, 1 2, 2) β solutions 16) β 17)( β 1, 2, 3) 18)( 1, 2, 2) β 19) (0, 2, 1) β 20) no solution 21) (10, 2, 3) Answers - Three Variables 1, 2) 3, 2) 12) solutions β 13) (0, 0, 0) 2) 3) 25) ( 26) ( 27) ( 28) ( 29) ( 1, β 3, 0) 1, β 3, 0) 8, 9) β β β ... |
5 23) 8 S20, 4 S10 24) 27 25) S12500 @ 12% S14500 @ 13% 26) S20000 @ 5% S30000 @ 7.5% 27) S2500 @ 10% S6500 @ 12% 28) S12400 @ 6% S5600 @ 9% 29) S4100 @ 9.5% S5900 @ 11% 30) S7000 @ 4.5% S9000 @ 6.5% 31) S1600 @ 4%; S2400 @ 8% 32) S3000 @ 4.6% S4500 @ 6.6% Answers - Mixture Problems 10) 1.5 11) 10 12) 8 13) 9.6 14) 36 ... |
34) 20 35) 35, 63 36) 3, 2 37) 1.2 38) 150 39) 10 40) 30, 20 41) 75 42) 20, 60 43) 25 Answers - Chapter 5 Answers to Exponent Properties 17) 42 18) 34 19) 3 20) 33 21) m2 22) xy3 4 23) 4x2y 3 24) y2 4 25) 4x10y14 26) 8u18v6 27) 2x17y16 28) 3uv 29) x2y 6 30) 4a2 3 31) 64 32) 2a 33) y3 512x24 34) y5x2 2 35) 64m12n12 36)... |
.815 20) 9.836 21) 5.541 22) 6.375 23) 3.025 24) 1.177 25) 2.887 26) 6.351 27) 2.405 6 10β 106 Γ Γ 3 10β Γ 108 104 1 10β Γ Γ Γ 5 10β 4 10β 9 10β 16 10β 6 10β 21 10β Γ Γ Γ Γ Γ Γ 20 10β Γ 10β 2 28) 2.91 Γ 458 31) 2y5x4 32) a3 2b3 33) 1 x2y11z 34) a2 8c10b12 35) 1 h3k j6 36) x30z6 16y4 37) 2b14 a12c7 38) m14q8 4p4 39) x2 ... |
) β 6 β 6 β 31) n3 32) 33) β β 5n2 + 3 β 6x4 + 13x3 12n4 + n2 + 7 34) 9x2 + 10x2 3r3 + 7r2 + 1 35) r4 β 36) 10x3 6x2 + 3x 8 β β 37) 9n4 + 2n3 + 6n2 38) 2b4 39) β b3 + 4b2 + 4b β 3b4 + 13b3 11b + 19 7b2 β β 40) 12n4 n3 6n2 + 10 β x3 β 4x + 2 41) 2x4 β 42) 3x4 + 9x2 + 4x β 5.5 Answers to Multiply Polynomials 1) 6p 42 β 2... |
6n + 36 β 30) 14a4 + 30a3 13a2 β 31) 15k4 + 24k3 + 48k2 + 27k + 18 β 12a + 3 33) 18x2 34) 10x2 35) 24x2 36) 16x2 37) 7x2 β 38) 40x2 12 15x β 55x + 60 18x 15 β β β β 44x 12 β β 49x + 70 10x 5 β β 39) 96x2 6 β 40) 36x2 + 108x + 81 5.6 1) x2 2) a2 3) 1 β 4) x2 64 β 16 β 9p2 9 β 49n2 25 64 64 β β 9 5) 1 β 6) 64m2 7) 25n2 ... |
3 9 + 5x2 + 4x 9 4) 3k2 8 + k 2 + 1 4 460 5) 2x3 + 4x2 + x 2 6) 5p3 4 + 4p2 + 4p 19) x 3 + 3 10x β β 20 33) 3n2 9n 10 β β 8 n + 6 β 5 9) x 7) n2 + 5n + 1 5 8) m2 3 + 2m + 3 10 + 9 x + 8 β 10 11) n + 8 9 β 12 13) v + 8 β 14) x 3 β β v 10 β 5 x + 7 15 16) x 6 x β β β 17) 5p + 4 + 3 4 9p + 4 18) 8k 9 β 1 β 3k 1 β 6.1 1) 9... |
6 Answers - Greatest Common Factor 9) 3a2b( 1 + 2ab) β 10) 4x3(2y2 + 1) 11) 5x2(1 + x + 3x2) β 12) 8n5( β 13) 10(2x4 4n4 + 4n + 5) 3x + 3) β 14) 3(7p6 + 10p2 + 9) 15) 4(7m4 + 10m3 + 2) 461 16) 2x( β 5x3 + 10x + 6) 17) 5(6b9 + ab 3a2) β 18) 3y2(9y5 + 4x + 3) 19) β 8a2b (6b + 7a + 7a3) 20) 5(6m6 + 3mn2 5) β 21) 5x3y2z(4... |
) β 5 + 3y3 β 2xy3 4xy) β β 32) 2y7( Answers - Grouping 11) (7x + 5)(y β 12) (7r2 + 3)(6r 7) 21) (4u + 3)(8v 5) β 7) 22) 2(u + 3)(2v + 7u) β 13) (8x + 3)(4y + 5x) 14) (3a + b2)(5b 15) (8x + 1)(2y 2) 7) β β 8) 16) (m + 5)(3n β 17) (2x + 7y2)(y 4x) β 5)(5n + 2) y)(8y + 7) 1)(y + 7) 18) (m 19) (5x 20) (8x β β β 23) (5x + ... |
21) (x 22) (u 23) (x β β β 9y)(x 2y) β 7v)(u 2v) β 3y)(x + 4y) 24) (x + 5y)(x + 9y) 25) (x + 6y)(x 2y) β 26) 4(x + 7)(x + 6) 28) 5(n 29) 6(a 30) 5(v β β β 8)(n 1) β 4)(a + 8) 1)(v + 5) 31) 6(x + 2y)(x + y) 32) 5(m2 + 6mn 18n2) β 33) 6(x + 9y)(x + 7y) 34) 6(m β 9n)(m + 3n) 1) (7x 6)(x 6) β β 2) (7n 2)(n β 3) (7b + 1)(b... |
)(4r + 7) 31) (x + 2y)(4x + y) 32) 2(2m2 + 3mn + 3n2) 3n)(4m + 3n) 33) (m β 34) 2(2x2 3xy + 15y2) β 35) (x + 3y)(4x + y) 36) 3(3u + 4v)(2u 3v) β 37) 2(2x + 7y)(3x + 5y) 38) 4(x + 3y)(4x + 3y) 39) 4(x 2y)(6x y) β β 40) 2(3x + 2y)(2x + 7y) 14) (3r + 7)(r + 3) 28) (3p + 7)(2p 1) β 6.5 1) (r + 4)(r 2) (x + 3)(x 4) β 3) β A... |
v + 1)(2v 7) (3k + 2)(3k 8) (3a + 1)(3a 2) β β β 1) 2) 1) 3) 9) 3(x + 3)(x β 10) 5(n + 2)(n β 11) 4(2x + 3)(2x 2) 3) β 12) 5(25x2 + 9y2) 13) 2(3a + 5b)(3a 5b) β 14) 4(m2 + 16n2) 15) (a 1)2 β 16) (k + 2)2 17) (x + 3)2 4)2 3)2 2)2 18) (n 19) (x β β 20) (k β 21) (5p β 22) (x + 1)2 23) (5a + 3b)2 24) (x + 4y)2 25) (2a β 26... |
14) 5(3u 3y)(x 5v)2 β 5) y) β β 15) (3x + 5y)(3x 5y) β 3y)(x2 + 3xy + 9y2) 16) (x β 17) (m + 2n)(m 27) (3x 4)(9x2 + 12x + 16) β 28) (4a + 3b)(4a 29) x(5x + 2) 30) 2(x β 31) 3k(k 2)(x β 5)(k β β 32) 2(4x + 3y)(4x 3b) β 3) 4) 3y) β 4x)(n + 3) 2n) β 33) (m β 18) 3(2a + n)(2b β 19) 4(3b2 + 2x)(3c 3) 2d) β 20) 3m(m + 2n)(m... |
4, 7 20) 1 4 21) β 22) 8, β β 4 8 3 2 23) 8, β 24) 4, 0 465, 1 25) 8 3, 26) 27) 28) β β β 29) β 30) 2, 3 β 7, 7 β 6 8 7 4, 5 2 6 5,, β β β β β β 31) 32) 33) 34) 35) 4 5, 6 β 36) 5 3, 2 β Answers - Chapter 7 7.1 1) 3 2) 1 3 3) 1 5 β 4) undeο¬ned 5) 1 2 6) 6 10 7) β 8) 0, 2 5 2 9) β 10) 0, 10 β 11) 0 12) β 13) β 14) 0, 1... |
) 63 10m 466 5) 3x2 2 6) 5p 2 7) 5m 8) 7 10 9) r 6 β r + 10 10) x + 4 11) 2 3 12) 9 β b 5 13) x 14) v 10 10 β 7 1 β 15) x + 1 16) a + 10 a 6 β 17) 5 18) p β p β 10 4 7.3 1) 18 2) a2 3) ay 4) 20xy 5) 6a2c3 6) 12 7) 2x 8) x2 8 β 2x β x 9) x2 β 10) x2 β 3 β 12 β 11x + 30 19) 3 5 20) x + 10 x + 4 21) 4(m β 5m2 5) 22) 7 23)... |
x 6x + 9 β 3)(x + 2) β 24) x(x 5, 2x x(x 6) 3x 12 6), β x(x 6) β β β x2 4x 4)2(x + 4) β, 25) (x β (x 5x + 1 5)(x + 2) β x2 + 7x + 6 26) 27) β 3x2 + 12x 4)2(x + 4) (x β, (x 4x + 8 5) (x + 2) β 6)(x + 6)2, 2x2 (x 9x 18 β β 6)(x + 6)2 β (x β 28) (x 3x2 + 4x + 1 4)(x + 3)(x + 1) β, (x β 2x2 8x 4)(x + 3)(x + 1) β 4x 3)(x + ... |
.5 1) x 2) 1 x β β y 1 y 3) β a a + 2 4) β 6) b3 + 2b b 2 β β 8b 7) 2 5 8) 4 5 9) 1 2 β 10) β 11) x2 1 2 x 1 x2 + x + 1 β β β β 21) 4t 5 4(t 3) 22) 2x + 10 (x + 3)2 23) 24) 20x 6 β 15x(x + 1) 9a 4(a 5) β 25) t2 + 2ty y2 26) 2x2 β x(x y2 β t2 β 10x + 25 5) β 3 x 34) 2x + 7 x2 + 5x + 6 35) 2x β 5x x2 β 8 β 14 36) β 3x2 +... |
x2 + 1 β 2 y 29) β 30) x2 31) y β xy 1 β x 32) x2 β y xy + y2 x β 33) x2 + y2 xy 34) 2x 1 β 2x + 1 7.6 1) 40 3 = a 2) n = 14 3 3) k = 12 7 4) x = 16 5) x = 3 2 6) n = 34 7) m = 21 8) x = 79 8 9) p = 49 10) n = 25 40 3 11) b = β 12) r = 36 5 13) x = 5 2 14) n = 32 5 7.7 1) β 2) β 3) 3 4) β 5) 2 6) 1 3 1, 2 2 3 3, 1 1, ... |
6.1 mi 6) 0.5 yd2 7) 0.435 km2 8) 86,067,200 ft2 9) 6,500,000 m3 10) 239.58 cm3 11) 0.0072 yd3 12) 5.13 ft/sec 13) 6.31 mph 14) 104.32 mi/hr 15) 111 m/s 8.1 1) 7 5β 2) 5 5β 26) 1 2 27) 3 10 28) 1 29) 30) 2 3 1 β β 31) 13 4 32) 1 10 33) β 34) 7 4 Answers - Dimensional Analysis 16) 2,623,269,600 km/yr 17) 11.6 lb/in2 18... |
2 7β 56x2 16 2nβ β β β β 23) 30 mβ β 24) 32p 7β 25) 3xy 5β 26) 6b2a 2aβ 27) 4xy xyβ 28) 16a2b 2β 29) 8x2y2 5β 30) 16m2n 2nβ Answers - Higher Roots 5β 15) 2 7n3 5β 2 3x3 16) β 17) 2p 75β 18) 2x 46β 19) β 20) β 21) 4v2 7β 6 7r 7β 16b 3b 3β 6v 22) 20a2 23β 23) 24) β β 25) 28n2 53β 8n2 3β 3xy 5x2 β 26) 4uv u2 3β 27) β 3β 2... |
3β β 3 2β + 6 5β 3β β 5 6β 5 6β 3 3β 2) 3) 4) 5) 6) β β β β β 7) 3 6β + 5 5β 8) 9) 10) 11) 12) 5β + 3β 8 2β β β 6 6β + 9 3β 3 6β + 3β 2 5β 6 6β β β β β 2 2β 13) β 14) 8 5β 3β β 15) 5 2β 16) 17) 9 3β 3 6β β β 3β β 18) 3 2β + 3 6β 12 2β + 2 5β 3 2β 19) 20) 8.4 β β 48 5β 1) β 2) 25 6β β 3) 6m 5β 25r2 2rβ 4) β 5β 22) β 23... |
β 9 5vβ β 13) β 14) 16 2 4 2β β 9 3β β 15) 15 11 5β β 16) 30 + 8 3β + 5 15β + 4 5β 17) 6a + a 10β + 6a 6β + 2a 15β 18) β 4p 10β + 50 pβ 19) 63 + 32 3β 10 mβ + 25 2β + 2mβ 5 β 20) 21) β 3β 25 25) 26) 15β 3 10β 15 27) 4 3β 9 28) 4 5β 5 29) 5 3xyβ 12y2 30) 4 3xβ 16xy2 31) 6pβ 3 32) 2 5nβ 5 33) 34) 35) 36) 103β 5 153β 4 10... |
2β β 30 10 + 5 10β 37) β 5 2β + 10 β 5 3β + 6β 38) 8 + 3 6β 10 18) 16 3β + 4 5β 43 19) 2β 1 β 20) 3 + 2 3β 21) 2β 22) 2β 23) aβ 24) 3 2 2β β aβ 25) 26) 1 3 27) 4 β 28) 2 5β 8.6 2 3β + 2 6β 3 2β β β 2 15β + 3β + 3 2 β 1) ( m5β )3 2) 1 4β )3 ( 10r 3) ( 7xβ )3 4) 1 3β )4 ( 6b 5) (6x)β 3 2 1 2 6) v 7 4 7) nβ 8) (5a) 1 2 A... |
y5z5 p17) 6 x3(x 2)2 β 3x(y + 4)2 p18) 4 p19) 10 x9y7 10β p20) 4a9b9 21) 12 x11y10 20β p22) a18b17 23) 20β a18b17c14 Answers - Radicals of Mixed Index 24) 30 x22y11z27 p25) a a4β 26) x xβ 27) b b9 10β 28) a a5 12β 6 29) xy xy5 10β 30) a ab7 p 31) 3a2b ab 4β 32) 2xy2 6 2x5y 33) x 12 p 59049xy11z10 34) a2b2c2 p 6β a2b c2... |
i β 6 30) 4i + 2 3 31) 3i 6 β 4 32) 5i + 9 9 33) 10i + 1 2i 34) β 35) β 40i + 4 101 36) 9i 45 β 26 37) 56 + 48i 85 6i 38) 4 β 13 39) 70 + 49i 149 40) β 36 + 27i 50 41) β 5 30i 37 β 42) 48i β 85 56 43) 9i 44) 3i 5β 45) 2 5β β 2 6β 46) β 47) 1 + i 3β 2 48) 2 + i 2β 2 i 49) 2 β 50) 3 + 2i 2β 2 51) i i 52) β 53) 1 54) 1 55... |
11) 65, 12) 5 63 β 13) β 14) β 15) 11, 2 16) 17) β β 7 11 2, 5 2 5 2 β 191 64 3 8, β 5 8 18) 9 8 19) 5 4 20) No Solution 34 3, 10 β 21) β 22) 3 23) 17 2 β 24) No Solutoin Answers - Complete the Square 17) β 5 + 86β, 18) 8 + 2 29β, 8 β β 19) 9, 7 86β 5 β 2 29β β 1 + i 21β, 20) 9, 1 21) β 22) 1, 23) 3 2, 24) 3, β β 3 7 ... |
9 + 21β 2, 9 β 21β 2 50) 1 + i 163β 2, 1 β i163 2 51) β 5 + i 415β 8, β 5 β i 415β 8 52) 11 + i 95β 6, 11 β i 95β 6 53) 5 + i 191β 2, 5 β i 191β 2 54) 8, 7 55) 1, 56) 3, 5 2 3 2 β β Answers - Quadratic Formula 3 + 141β 6 15) β, β 3 141β β 6 5β 17) β 16) 3β, 3β β 3 + 401β 14 3, β 401β β 14 3, 2) i 6β i 6β 3 3) 2 + 5β, ... |
i 143β 14 37) β 3 + 345β 14 3, β 345β β 14 38) 6β 2, 39) 26β 2 β, 6β 2 26β 2 β 1 + 141β 10 1, β 141β β 10, 2 β 3i 5β 7 40) β 9.5 Answers - Build Quadratics from Roots NOTE: There are multiple answers for each problem. Try checking your answers because your answer may also be correct. 14x + 13 = 0 22x + 40 = 0 β β β 7x... |
6 1) 2) 3) 4) 1, 2, i, 5β 2 2β 2 Answers - Quadratic in Form 5) 6) 7) 8) 1, 3, 3, 6, 7 1 4 2 Β± Β± Β± Β± Β± Β± Β± Β± 480 9) 2, 4 Β± Β± 10) 2, 3, 1 i 3β, β 3 3i 3β Β± 2 Β± i 3β, β i 3β 3 Β± 2 β 2, 3, 1 6β, Β± 2i 6β 2 Β±, Β± 2i 3β 3 1 3 β 125, 343 5 4, 1 5 11) β 12) Β± 13) Β± 14) 1 4, 15) β 16) β 17) 1, Β± Β± Β± Β± Β± 18) 19) 20) 21) 22) 23), ... |
β Β± 46) 1, 2, 1, 3 1 2, β 2, 1 3 1 β Answers - Rectangles 10) 1.54 in 11) 3 in 12) 10 ft 13) 1.5 yd 14) 6 m x 8 m 15) 7 x 9 16) 1 in 17) 10 rods 18) 2 in 481 19) 15 ft 20) 20 ft 21) 1.25 in 22) 23.16 ft 23) 17.5 ft 24) 25 ft 25) 3 ft 26) 1.145 in 9.8 1) 4 and 6 2) 6 hours 3) 2 and 3 4) 2.4 5) C = 4, J = 12 6) 1.28 day... |
) r = 5 10) 60, 80 11) 6 km/hr 12) 200 km/hr 13) 56, 76 14) 3.033 km/hr 15) 12 mph, 24 mph 18) 36 mph 19) 45 mph 20) 40 mph, 60 mph 21) 20 mph 16) 30 mph, 40 mph 22) 4 mph 482 1) (-2,0) (4,0) (0,-8) (1,-9) 2) (-1,0) (3, 0) (0, -3) (1, -4) 3) (0,10) (1,0) (5,0) (3,-8) 4) 5) 6) (0,16) (2,0) (0,-18) (1,0) (-10,0) (4,0) (3... |
.1 1) a. yes b. yes c. no d. no e. yes f. no g. yes h. no 2) all real numbers 3) x 6 5 4 4) t 0 5) all real numbers 6) all real numbers 1, 4 7) x > 16 8) x β 9) x > 4, x 5 10) x Β± 4 11) β 5 3 25 12) β 13) 2 10.2 1) 82 2) 20 3) 46 4) 2 5) 5 Answers - Function Notation 14) 85 7 15) β 16) 7 17 9 6 21 17) β 18) β 19) 13 20... |
2x β 3x + 4 37) β β 38) x4 39) β 4x2 3 β β n2 β 3 2n 40) 32 + 23n 8 β n3 155 41) β 42) 5 43) 21 44) 4 45) 103 46) 12 47) 50 β 48) 112 49) 176 50) 147 51) 16x2 + 12x 4 β 8a + 14 8a + 2 52) 53) β β 54) t 55) 4x3 12n 16 β 2n2 β 2x + 8 56) β 57) β 58) 27t3 β 16t 5 59) β β 60) 3x3 + 6x2 4 β 108t2 + 141t 60 β Answers - Inve... |
β 1(x) = 5 + 4x 5 31) gβ 3β 1(x) = x + 1 32) f β 1(x) = β 5 x + 3 2 33) hβ 34) gβ q β 1(x) = ( 2x + 4)3 4 2 + 1 3β 1(x) = x β 2x + 1 1(x) = β x 1 β 36) f β 1(x) = β x 1 β x 37) f β 1(x) = 2x + 7 x + 3 4x 3 x 38) f β 1(x) = β 39) gβ 40) gβ 1(x) = β 1(x) = β 3x + 1 2 35) f β 3 Answers - Exponential Functions 15) 1 16) 1... |
; 953.44 e. 1209.52; 1214.87 f. 1528.02; 1535.27 g. 2694.70; 2699.72 d. 1979.22; 1984.69 h. 3219.23; 3224.99 i. 7152.17; 7190.52 2) 1640.70 3) 2868.41 4) 2227.41 5) 1726.16 6) 1507.08 7) 2001.60 8) 2009.66 9) 2288.98 10) 6386.12 11) 13742.19 487 12) 28240.43 13) 12.02; 3.96 14) 3823.98 15) 101.68 10.7 1) 0.3256 2) 0.92... |
55β¦ 26) 30.5β¦ 27) 47β¦ 28) 15.5β¦ 29) 30β¦ 30) 59β¦ 32) mβ B = 22.8β¦, mβ A = 67.2β¦, c = 16.3 33) mβ B = 22.5β¦, mβ A = 67.5β¦, c = 7.6 34) mβ A = 39β¦, b = 7.2, a = 5.9 35) mβ B = 64.6β¦, mβ A = 25.4β¦, b = 6.3 36) mβ A = 69β¦, b = 2.5, a = 6.5 37) mβ B = 38β¦, b = 9.9, a = 12.6 38) mβ B = 42β¦, b = 9.4, c = 14 39) mβ A = 45β¦, b = 8, c = 11... |
calculated so that area β frequency. The vertical axis of the histogram is labelled frequency density, which measures frequency per standard interval. The simplest and most commonly used standard interval is 1 unit of measurement. For example, a column representing 85 objects with masses from 50 to 60 kg has a frequen... |
Our estimate for the interval 50β63kg is equal to the area corresponding to this section of the second column. We add together the estimates for the two intervals. Copyright Material - Review Only - Not for Redistribution TIP Column areas are equal to class frequencies. For example, the area of the first column is 4 c... |
values with much greater accuracy than scales such as 1cm for 3, 7 or 23 units. For similar reasons, try to use as much of the sheet of graph paper as possible, ensuring that the whole diagram will fit before you start to draw it. 9 We can see that two blocks represent one athlete in the histogram. So, instead of calc... |
the table. b On graph paper, draw a histogram to represent the data. c Calculate an estimate of the number of samples with masses between 8 and 18 grams. 3 The table below shows the heights, in metres, of 50 boys and of 50 girls. Height (m) No. boys f( ) No. girls f( ) 1.2β 7 10 1.3β 11 22 1.6β 26 16 1.8β1.9 6 2 a How... |
given that 24 of these journeys were delayed by less than 2 minutes 20 16 12 8 4 0 2 4 12 Time (min) 18 20 11 a How many journeys were monitored? b Calculate an estimate of the number of these journeys that were delayed by: i 1 to 3 minutes ii 10 to 15 minutes. c Show that a total of 2160 journeys were provided in Aug... |
, medium or thick in the ratio 1: 3 : 1. Estimate the thickness of a medium sheet, giving your answer in the form < < your values for a and b? a k b. How accurate are PS 10 The masses, in kilograms, of the animals treated at a veterinary clinic in the past year are illustrated in a histogram. The histogram has four col... |
The Commercial and Political Atlas (London, 1786). His invention was adopted by many in the following years, including Florence Nightingale, who used bar charts in 1859 to compare mortality in the peacetime army with the mortality of civilians. This helped to convince the government to improve army hygiene. In the pas... |
Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: P... |
, which has a cumulative frequency of 0, and plot this point on your graph. β’ The dotted lines show the workings for parts a and b. β’ When constructing a cumulative frequency graph, you are advised to use sensible scales that allow you to plot and read values accurately. β’ Estimates from a polygon and a curve will not ... |
Draw a cumulative frequency polygon to represent the data. b Use your graph to estimate: i the number of participants with reaction times between 5.5 and 7.5 seconds ii the lower boundary of the slowest 20 reaction times. 2 The following table shows the widths of the 70 books in one section of a library, given to the ... |
an estimate. b Estimate the number of components that have: i a diameter of less than 0.15cm ii a radius of 0.16cm or more. c Estimate the value of k, given that 20% of the components have diameters of k mm or more. d Give the reason why 0.1β0.2 cm is the modal class. 17 5 The following cumulative frequency graph show... |
many staff take between 15 and 45 minutes to get to work? b Find the exact number of staff who take + x y 2 minutes or more to get to work, given that 85% of the staff take less than x minutes and that 70% of the staff take y minutes or more. M 8 A fashion company selected 100 12-year-old boys and 100 12-year-old girl... |
6β 7 8.8β 52 12.1β 12 15.4β18.7 4 These distances are 10% greater than the distances the cars will be able to travel after they have covered more than 100 000 km. Estimate how many of the cars can travel 10.5km or more on 1 litre of fuel when new, but not after they have covered more than 100 000 km. 19 PSM 11 A small ... |
six plotted points, whose coordinates you know, are joined by straight lines. b Arranged in ascending order, the mass of the 100th object 50 g. = However, we can calculate these estimates from the information given in the table without drawing the polygon. Investigate the possible methods that we can use to calculate ... |
their favourite colour. a List the methods of representation that would be suitable for displaying Jamilaβs data. b Jamila wishes to emphasise that the favourite colour of exactly three-quarters of the students is blue. Which type of representation from your list do you think would be the most effective for Jamila to ... |
Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 6 The percentage score... |
Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 1: Representa... |
: Probability & Statistics 1 END-OF-CHAPTER REVIEW EXERCISE 1 1 The weights of 220 sausages are summarised in the following table. Weight (grams) Cumulative frequency 20< 0 30< 20 40< 50 45< 100 50< 160 60< 210 70< 220 i State how many sausages weighed between 50 g and 60 g. ii On graph paper, draw a histogram to repre... |
min) No. downloads cf( ) 3< 6 5< 18 6< 66 10< 80 a Find the number of downloads that completed in 5 to 6 minutes. b On a histogram, the download times from 5 to 6 minutes are represented by a column of height 9.6cm. Find the height of the column that represents the download times of 6 to 10 minutes. [1] [2] PS 7 A hist... |
when more than 30 rooms were occupied. iii On 75% of the days at most n rooms were occupied. Estimate the value of n. [4] [2] [2] Cambridge International AS & A Level Mathematics 9709 Paper 62 Q5 June 2011 Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy ... |
of all the people in a region, then a single value (i.e. an average income) would be a convenient number to represent our findings. However, choosing which average to use is something that needs to be thought about, as one measure may be more appropriate to use than the others. Deciding which measure to use depends on... |
PLE 2.1 28 Find the modal class of the 270 pencil lengths, given to the nearest centimetre in the following table. Length x( cm) No. pencils ( )f 4β7 8β10 11β12 100 90 80 Answer Length ( cm) x No. pencils ( )f 3.5 < <x 7.5 7.5 < <x 10.5 10.5 < <x 12.5 100 90 80 Width (cm) Frequency density 4 3 2 100 Γ· = 4 25 90 3 30 Γ· ... |
equal, so class frequencies are in the same ratio as interval widths. Alternatively, frequencies are proportional to interval widths. TIP In the special case where all classes have equal widths, frequency densities are proportional to frequencies, so the modal class is the class with the highest frequency. 1 Find the ... |
classes is 120. Find the least possible frequency of the modal class, given that the modal class is 4β10. 2.2 The mean The mean is referred to more precisely as the arithmetic mean and it is the most commonly known average. The sum of a set of data values can be found from the mean. Suppose, for example, that 12 value... |
οΏ½ f Ξ£. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University ... |
answer assumes that the masses given are accurate to 1 decimal place; that the numbers of sweets given are accurate; and that the masses of the bags are not included in the given totals. Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridg... |
not including 25kg. 22 crates contain coconuts with a total mass from 25 up to but not including 40 kg. 7 crates contain coconuts with a total mass from 40 up to but not including 54kg. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambri... |
yearolds are all aged from 20 up to but not including 22 years. The age groups and necessary totals are shown in the table. Copyright Material - Review Only - Not for Redistribution TIP Incorrect mid-values of 18, 19 and 20.5 give an incorrect estimated mean of 19.1 years. FAST FORWARD We will see how the mean is used ... |
x 8 8 << x 14 8 9 11 2 13 << y 16 16 << y 21 21 << y 28 28 << y 33 33 << y 36 7 17 29 16 11 7 An examination was taken by 50 students. The 22 boys scored a mean of 71% and the girls scored a mean of 76%. Find the mean score of all the students. 8 A company employs 12 drivers. Their mean monthly salary is $1950. A new ... |
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