169_186_CC_A_RSPC1_C12_662330.indd



Chapter 6 Review6-1 - Operations on FunctionsArithmetic OperationsOperations with FunctionsSum(f + g)(x) = f (x) + g(x)Difference(f – g)(x) = f (x) – g(x)Product(f ? g)(x) = f (x) ? g(x)Quotientfg(x) = f(x)g(x) , g(x) ≠ 0Example : Find (f + g)(x), (f – g)(x), (f ? g)(x), and fg (x) for f(x) = x2 + 3x – 4 and g(x) = 3x – 2.(f + g)(x) = f(x) + g(x) Addition of functions = (x2 + 3x – 4) + (3x – 2) f(x) = x2 + 3x – 4, g(x) = 3x – 2 = x2 + 6x – 6 Simplify. (f – g)(x) = f(x) – g(x) Subtraction of functions = (x2 + 3x – 4) – (3x – 2) f(x) = x2 + 3x – 4, g(x) = 3x – 2 = x2 – 2 Simplify. (f ? g)(x) = f(x) ? g(x) Multiplication of functions = (x2 + 3x – 4)(3x – 2) f(x) = x2+ 3x – 4, g(x) = 3x – 2 = x2(3x – 2) + 3x(3x – 2) – 4(3x – 2) Distributive Property = 3x3 – 2x2 + 9x2 – 6x – 12x + 8 Distributive Property = 3x3 + 7x2 – 18x + 8 Simplify.fg(x) = f(x)g(x) Division of functions = x2+ 3x - 43x - 2 , x ≠ 23 f(x) = x2 + 3x – 4 and g(x) = 3x – 2ExercisesFind (f + g)(x), (f – g)(x), (f ? g)(x), and fg (x) for each f(x) and g(x).1. f(x) = 8x – 3; g(x) = 4x + 5 2. f(x) = x2 + x – 6; g(x) = x – 23. f(x) = 3x2 – x + 5; g(x) = 2x – 3 4. f(x) = 2x – 1; g(x) = 3x2 + 11x – 46-1 Continued - Operations on FunctionsComposition of Functions Suppose f and g are functions such that the range of g is a subset of the domain of f. Then the composite function f ? g can be described by the equation [f ° g](x) = f[g(x)].Example 1: For f = {(1, 2), (3, 3), (2, 4), (4, 1)} and g = {(1, 3), (3, 4), (2, 2), (4, 1)}, find f ? g and g ? f if they exist.f[ g(1)] = f(3) = 3 f[ g(2)] = f(2) = 4 f[ g(3)] = f(4) = 1 f[ g(4)] = f(1) = 2,So f ? g = {(1, 3), (2, 4), (3, 1), (4, 2)}g[ f(1)] = g(2) = 2 g[ f(2)] = g(4) = 1 g[ f(3)] = g(3) = 4 g[ f(4)] = g(1) = 3,So g ? f = {(1, 2), (2, 1), (3, 4), (4, 3)}Example2: Find [g ? h](x) and [h ? g](x) for g(x) = 3x – 4 and h(x) = x2 – 1.[g ? h](x) = g[h(x)] [h ? g](x) = h[ g(x)] = g(x2 – 1) = h(3x – 4) = 3(x2 – 1) – 4 = (3x-4)2 – 1 = 3x2 – 7 = 9x2 – 24x + 16 – 1 = 9x2 – 24x + 15ExercisesFor each pair of functions, find f ? g and g ? f , if they exist.1. f = {(–1, 2), (5, 6), (0, 9)}, 2. f = {(5, –2), (9, 8), (–4, 3), (0, 4)},g = {(6, 0), (2, –1), (9, 5)} g = {(3, 7), (–2, 6), (4, –2), (8, 10)}Find [f ? g](x) and [g ? f ](x), if they exist.3. f(x) = 2x + 7; g(x) = –5x – 1 4. f(x) = x2 – 1; g(x) = –4x25. f(x) = x2 + 2x; g(x) = x – 9 6. f(x) = 5x + 4; g(x) = 3 – x6-2 Study Guide - Inverse Functions and RelationsFind InversesInverse RelationsTwo relations are inverse relations if and only if whenever one relation contains the element (a, b), the other relation contains the element (b, a).Property of Inverse FunctionsSuppose f and f-1 are inverse functions. Then f (a) = b if and only if f-1(b) = a.Example: Find the inverse of the function f(x) = 25 x – 15. Then graph the function and its inverse.304015628127Step 1 Replace f(x) with y in the original equation.f(x) = 25 x – 15 → y = 25 x – 15Step 2 Interchange x and y.x = 25 y – 15Step 3 Solve for y.x = 25 y – 15Inverse of y = 25 x – 155x = 2y – 1 Multiply each side by 5.5x + 1 = 2y Add 1 to each side.12 (5x + 1) = y Divide each side by 2.The inverse of f(x) = 25 x – 15 is f-1(x) = 12 (5x + 1).ExercisesFind the inverse of each function. Then graph the function and its inverse.1. f(x) = 23 x – 1 2. f(x) = 2x – 3 3. f(x) = 14 x – 2240347531115471614531115180415315266-2 - Inverse Functions and Relations (continued)Verifying InversesInverse FunctionsTwo functions f(x) and g(x) are inverse functions if and only if [f ? g](x) = x and [g ? f ](x) = x.Example 1: Determine whether f(x) = 2x – 7 and g(x) = 12 (x + 7) are inverse functions.[ f ? g](x) = f[ g(x)] [g ? f ](x) = g[ f(x)]= f 12(x + 7) = g (2x – 7)= 2 12(x + 7) – 7 = 12 (2x – 7 + 7)= x + 7 – 7 = x= xThe functions are inverses since both [ f ? g](x) = x and [ g ? f ](x) = x.Example 2: Determine whether f(x) = 4x + 13 and g(x) = 14 x – 3 are inverse functions.[ f ? g](x) = fg(x) = f 14x – 3= 4 14x – 3 + 13= x – 12 + 13= x – 1123Since [ f ? g](x) ≠ x, the functions are not inverses.ExercisesDetermine whether each pair of functions are inverse functions. Write yes or no.1. f(x) = 3x – 1 2. f(x) = 14 x + 5 3. f(x) = 12 x – 10g(x) = 13 x + 13g(x) = 4x – 20 g(x) = 2x + 1104. f(x) = 2x + 55. f(x) = 8x – 12 6. f(x) = –2x + 3g(x) = 5x + 2 g(x) = 18 x + 12 g(x) = – 12 + 326-3 - Square Root Functions and InequalitiesSquare Root Functions A function that contains the square root of a variable expression is a square root function. The domain of a square root function is those values for which the radicand is greater than or equal to 0.Example: Graph y = 3x-2. State its domain and range.Since the radicand cannot be negative, the domain of the function is 3x – 2 ≥ 0 or x ≥ 23.The x-intercept is 23. The range is y ≥ 0.Make a table of values and graph the function.x1120775-48260y230112237ExercisesGraph each function. State the domain and range.1. y = 2x 2. y = –3 x 3. y = – x22436723-16004750539106163189171-18754. y = 2 x-3 5. y = – 2x-36. y = 2x+5244403910297247505381024422273301022356-3 - Square Root Functions and Inequalities(continued)Square Root Inequalities A square root inequality is an inequality that contains the square root of a variable expression. Use what you know about graphing square root functions and graphing inequalities to graph square root inequalities.4697376-4164Example: Graph y ≤ 2x-1 + 2.Graph the related equation y = 2x-1 + 2. Since the boundary should be included, the graph should be solid.The domain includes values for x ≥ 12, so the graph is to the right of x = 12.ExercisesGraph each inequality.1. y < 2x 2. y > x+33. y < 32x-12129031205744741012547372436723766834. y < 3x-45. y ≥ x+1 – 4 6. y > 22x-34733697931932485390-1905242334-19056-4 - nth RootsSimplify RadicalsSquare RootFor any real numbers a and b, if a2 = b, then a is a square root of b.nth RootFor any real numbers a and b, and any positive integer n, if an = b, then a is an nth root of b.Real nth Roots of b,nb, – nb1. If n is even and b > 0, then b has one positive real root and one real negative root.2. If n is odd and b > 0, then b has one positive real root.3. If n is even and b < 0, then b has no real roots.4. If n is odd and b < 0, then b has one negative real root.Example 1: Simplify 49z8.49Z8 = (7z4)2 = 7z4z4 must be positive, so there is no need to take the absolute value.Example 2: Simplify – 3(2a-1)6 – 32a-16= 3[2a-12]3 = – 2a-12ExercisesSimplify.1.812.3-343 3. 144p64.±4a105.5243p106. – 3m6n97.3-b128.16a10b89. 121x610.(4k)411. ±169r412. – 3-27p613. – 625y2z414. 36q3415. 100x2y4z26-4 - nth Roots(continued)Approximate Radicals with a CalculatorIrrational Numbera number that cannot be expressed as a terminating or a repeating decimalRadicals such as 2 and 3 are examples of irrational numbers. Decimal approximations for irrational numbers are often used in applications. These approximations can be easily found with a calculator.Example: Use a calculator to approximate 518.2 to three decimal places. 318.2 ≈ 1.787ExercisesUse a calculator to approximate each value to three decimal places.1. 622. 10503. 30.0544. – 45.455. 52806. 18,6007. 0.0958. 3-159. 510010. LAW ENFORCEMENT The formula r = 2 5L is used by police to estimate the speed r in miles per hour of a car if the length L of the car’s skid mark is measures in feet. Estimate to the nearest tenth of a mile per hour the speed of a car that leaves a skid mark 300 feet long.11. SPACE TRAVEL The distance to the horizon d miles from a satellite orbiting h miles above Earth can be approximated by d = 8000h+h2. What is the distance to the horizon if a satellite is orbiting 150 miles above Earth?6-5 - Operations with Radical ExpressionsSimplify RadicalsProduct Property of RadicalsFor any real numbers a and b, and any integer n > 1:1. if n is even and a and b are both nonnegative, then nab = na ? nb.2. if n is odd, then nab = na ? nb.To simplify a square root, follow these steps:1. Factor the radicand into as many squares as possible.2. Use the Product Property to isolate the perfect squares.3. Simplify each radical.Quotient Property of RadicalsFor any real numbers a and b ≠ 0, and any integer n > 1,nab = nanb , if all roots are defined.To eliminate radicals from a denominator or fractions from a radicand, multiply the numerator and denominator by a quantity so that the radicand has an exact root.Example 1: Simplify 3-6a5b7.3-16a5b7 = (-2)3?2?a3?a2?(b2)3?b= – 2ab2 32a2bExample 2: Simplify 8x345y58x345y5 = 8x345y5Quotient Property= (2x)2? 2x(3y2)2 ? 5y Factor into squares.= (2x)2 ? 2x(3y2)2 ? 5y Product Property= 2x2x3y25ySimplify.= 2x2x3y25y ?5y5yRationalize the denominator.= 2x10xy15y3Simplify.ExercisesSimplify.1. 5542.432a9b203. 75x4y74.361255.a6b3986. 3p5q3406-5 - Operations with Radical Expressions(continued)Operations with Radicals When you add expressions containing radicals, you can add only like terms or like radical expressions. Two radical expressions are called like radical expressions if both the indices and the radicands are alike.To multiply radicals, use the Product and Quotient Properties. For products of the form (ab + c d) ? (e f + g h), use the FOIL method. Example 1: Simplify 250 + 4500 – 6125.250 + 4500 – 6125 = 252?2 + 4102?5 – 652?5 Factor using squares.= 2 ? 5 ? 2 + 4 ? 10 ? 5 – 6 ? 5 ? 5Simplify square roots.= 102 + 40 + 5 – 305Multiply.= 102 + 105Combine like radicals.Example 2: Simplify (23 – 42 ) (3+ 22) .(23 – 42 ) (3 + 22)= 2 3?3 + 23 ? 22 – 42?3 – 42 ? 22= 6 + 46 – 46 – 16= –10Example 3: Simplify 2 - 53 + 5 .2 - 53 + 5 = 2 - 53 + 5 ? 3 - 53- 5= 6 - 25 - 35 + (5)232- (5)2= 6 - 55 + 59 - 5= 11- 554ExercisesSimplify.1. 32 + 50 – 482. 20 + 125 – 453. 300 – 27 – 754.381 ?3245. 3234 +3126. 23 (15 + 60)7. (2 + 37) (4 + 7) 8. (63 – 42) (33 + 2) 9. (42 – 35) (220 + 5)6-6 - Rational ExponentsRational Exponents and RadicalsDefinition of b1nFor any real number b and any positive integer n, b1n = nb, except when b < 0 and n is even. Definition of bmnFor any nonzero real number b, and any integers m and n, with n > 1, bmn = nbm = nbm, except when b < 0 and n is even.Example 1: Write 2812 in radical form.Notice that 28 > 0. 2812 = 28= 22?7 = 22 ? 7= 27Example 2: Evaluate -8-12513.Notice that –8 < 0, –125 < 0, and 3 is odd.-8-12513 = 3-83-125= -2-5= 25ExercisesWrite each expression in radical form, or write each radical in exponential form.1. 11172. 1513 3. 300324. 475. 33a5b26. 4162p5Evaluate each expression.7. -27238. 216139. (0.0004)126-6 - Rational Exponents(continued)Simplify Expressions All the properties of powers from Lesson 6-1 apply to rational exponents. When you simplify expressions with rational exponents, leave the exponent in rational form, and write the expression with all positive exponents. Any exponents in the denominator must be positive integers.When you simplify radical expressions, you may use rational exponents to simplify, but your answer should be in radical form. Use the smallest index possible.Example 1: Simplify y23 ? y38.y23 ? y38 = y23 + 38 = y2524Example 2: Simplify 4144x6.4144x6 = 144x614= 24? 32?x614= 2414 ? 3214 ? x614= 2 ? 312 ? x32 = 2x ? 3x12 = 2x3xExercisesSimplify each expression.1. x45 ? x652. y23343. p45 ? p7104. m6525 5. x38 ? x436. s16437. pp138. x12x13 9. 6128 SKIP10. 449 SKIP11. a3b4ab312. 32 ? 316 TRY6-7 - Solving Radical Equations and InequalitiesSolve Radical Equations The following steps are used in solving equations that have variables in the radicand. Some algebraic procedures may be needed before you use these steps.Step 1 Isolate the radical on one side of the equation.Step 2 To eliminate the radical, raise each side of the equation to a power equal to the index of the radical.Step 3 Solve the resulting equation.Step 4 Check your solution in the original equation to make sure that you have not obtained any extraneous roots.Example 1: Solve 24x+8 – 4 = 8.24x+8 – 4 = 8Original equation24x+8 = 12Add 4 to each side.4x+8 = 6Isolate the radical.4x + 8 = 36Square each side.4x = 28Subtract 8 from each side.x = 7Divide each side by 4.Check24(7)+8 – 4 ? 8236 – 4 ? 82(6) – 4 ? 88 = 8The solution x = 7 checks.Example 2: Solve 3x+1 = 5x – 1. 3x+1 = 5x – 1Original equation3x + 1 = 5x – 25x + 1Square each side.25x = 2xSimplify.5x = xIsolate the radical.5x = x2Square each side.x2 – 5x = 0Subtract 5x from each side.x(x – 5) = 0Factor.x = 0 or x = 5Check3(0)+1 = 1, but 5(0) – 1 = –1, so 0 is not a solution.3(5)+1 = 4, and 5(5) – 1 = 4, so the solution is x = 5.ExercisesSolve each equation.1. 3 + 2x3 = 52. 23x+4 + 1 = 153. 8 + x+1 = 24. 5-x – 4 = 65. 12 + 2x-1 = 46. (9x-11)12 = x + 17. 432x+11 – 2 = 108. 10 – 2x = 59. 4 + 7x = 7x-96-7 - Solving Radical Equations and Inequalities(continued)Solve Radical Inequalities A radical inequality is an inequality that has a variable in a radicand. Use the following steps to solve radical inequalities.Step 1 If the index of the root is even, identify the values of the variable for which the radicand is nonnegative.Step 2 Solve the inequality algebraically.Step 3 Test values to check your solution.Example: Solve 5 – 20x+4 ≥ –3.Since the radicand of a square root must be greater than or equal to zero, first solve20x + 4 ≥ 0.20x + 4 ≥ 020x ≥ –4x ≥ – 15Now solve 5 – 20x+4 ≥ – 3.5 – 20x+4 ≥ –3Original inequality20x+4 ≤ 8Isolate the radical.20x + 4 ≤ 64Eliminate the radical by squaring each side.20x ≤ 60Subtract 4 from each side.x ≤ 3Divide each side by 20.ExercisesSolve each inequality.1. c-2 + 4 ≥ 72. 32x-1 + 6 < 153. 10x+9 – 2 > 54. 8 – 3x+4 ≥ 35. 2x+8 – 4 > 26. 9 – 6x+3 ≥ 6 ................
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