Exam Review 2 nd Semester 6-1 Operations on Functions

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1 NAME DATE PERIOD Exam Review 2 nd Semester 6-1 Operations on Functions Find (f + g)(x), (f g)(x), (f g)(x), and (x) for each f(x) and g(x). 1. f(x) = 8x 3; g(x) = 4x f(x) = + x 6; g(x) = x 2 If f(x) =, g(x) = 5x, and h(x) = x + 4, find each value. 3. f[ g(1)] 4. g[h(2)] 5. h[f(4)] Find [f g](x) and [g f ](x), if they exist. 6. f(x) = 2x + 7; g(x) = 5x 1 7. f(x) = 1; g(x) = Inverse Functions and Relations Find the inverse of each function. Then graph the function and its inverse. 1. f(x) = x 1 2. f(x) = 2x 3 3. f(x) = x 2

2 Determine whether each pair of functions are inverse functions. Write yes or no. 4. f(x) = 2x f(x) = 8x f(x) = 2x + 3 g(x) = 5x + 2 g(x) = x + 12 g(x) = Square Root Functions and Inequalities Graph each function. 1. y = 2. y = 3 3. y = Graph the following inequalities. 4. y 2 5. y y < nth Roots Simplify ±

3 Use a calculator to approximate each value to three decimal places Operations with Radical Expressions Simplify ( + ) 11. (2 + 3 ) (4 + ) Rational Exponents Write each expression in radical form, or write each radical in exponential form Evaluate each expression

4 Simplify each expression Solving Radical Equations and Inequalities Solve each equation x = = = = = 12. = x + 1 Solve each inequality < > >

5 12-1Trigonometric Functions in Right Triangles Find the values of the six trigonometric functions for angle θ In a right triangle, A and B are acute. 4. If tan B = 2, what is cos B? 5. If tan A =, what is sin A? 6. If sin B =, what is cos B? Use a trigonometric function to find each value of x. Round to the nearest tenth if necessary Use trigonometric functions to find the values of x and y. Round to the nearest tenth if necessary SURVEYING John stands 150 meters from a water tower and sights the top at an angle of elevation of 36. If John s eyes are 2 meters above the ground, how tall is the tower? Round to the nearest meter.

6 12-2 Angles and Angle Measure Draw an angle with the given measure in standard position Find an angle with a positive measure and an angle with a negative measure that are coterminal with each angle Rewrite each degree measure in radians and each radian measure in degrees π

7 12-3 Trigonometric Functions of General Angles The terminal side of θ in standard position contains each point. Find the exact values of the six trigonometric functions of θ. 1. (6, 8) 2. (20, 21) 3. (2, 5) Sketch each angle. Then find its reference angle Find the exact value of each trigonometric function. 7. tan cot cot (90 ) 10. cos tan 12. csc 13. cot 2π 14. tan

8 12-6Circular Functions The terminal side of angle θ in standard position intersects the unit circle at each point P. Find cos θ and sin θ P (0, 1) 5. P (1, 0) P(0.8, 0.6 ) 4. P (0, 1) Find the exact value of each function. 12. sin (390 ) 13. sin 5π 14. cos 3π 15. sin 16. sin 17. cos

9 12-7 Graphing Trigonometric Functions Name Find the amplitude and period of each function. Then graph the function. 1. y = 2 cos θ 2. y = 4 sin θ 3. y = sin 3θ 4. y = cos 2θ 5. y = cos 2θ 6. y = 4 sin θ 12-7 Translations of Trigonometric Graphs State the amplitude, period, and phase shift for each function. Then graph the function. 1. y = sin (θ + 90 ) y = cos (θ 45 ) State the amplitude, period, phase shift, and vertical shift of each function. Then graph the function. 3. y = 2 cos [3(θ + 45 )] y = 3 sin [2(θ 90 )] + 2

10 13-1 Trigonometric Identities Find the exact value of each expression if 0 < θ < If tan θ = 1, find sec θ. 2. If tan θ =, find cos θ. Find the exact value of each expression if 90 < θ < If cos θ =, find sin θ. 4. If cot θ =, find cos θ. Find the exact value of each expression if 180 < θ < If tan θ = 1, find cos θ. 6. If sin θ =, find tan θ. 7. If csc θ = 2, find cot θ. 8. If sin θ =, find tan θ. Simplify each expression. 9. sin θ sec θ 10. csc θ sin θ 11. cot θ sec θ tan θ + cot θ 14. csc θ tan θ tan θ sin θ csc θ + cot θ

11 8-1 Multiplying and Dividing Rational Expressions Simplify each expression Adding and Subtracting Rational Expressions Simplify each expression

12 8-6 Solving Rational Equations and Inequalities Solve each equation. Check your solution. 1. = 2 2. = 1 3. = 4. + = 4 5. = 6. + = 10 Solve each inequality. Check your solutions. 7. > 8. > 0 9. < 10. < 1

13 8-3 Graphing Reciprocal Functions Identify the asymptotes, domain, and range of each function. 1. f (x) = 2. f (x) = 3. f(x) = + 2 Graph each function. State the domain and range. 4. f(x) = f(x) = f(x) = Graphing Rational Functions Graph each function. 1. f(x) = 2. f(x) = 3. f(x) =

14 7-1Graphing Exponential Functions Graph each function. State the domain and range. Determine whether the graph represents growth or decay. 1. y = 2. y = 2 3. y = 7-2 Solving Exponential Equations and Inequalities Solve each equation. 1. = 2. = 3. = 4. = 5. = 6. = Write an exponential function for the graph that passes through the given points. 7. (0, 4) and (2, 36) 8. (0, 6) and (1, 81) 9. (0, 5) and (6, 320) 10. (0, 3) and (3, 24) 11. (0, 12) and (4, 144) 12. (0, 9) and (2, 49)

15 7-3 Logarithmic Functions and Expressions Write each equation in exponential form = 2 2. = = Write each equation in logarithmic form. 4. = = 6. = Evaluate each expression , Describe the transformation of each parent function below. Then graph each function. 13. f(x) = x f(x) = 2 (x + 2) 15. f(x) = - (x -1) Solving Logarithmic Equations and Inequalities Solve each equation = 3x = 2 3. (5x + 1) = 2 4. (x 5) = 5. (3x 1) = (2x + 3) 6. x = (2x 1)

16 7-5 Properties of Logarithms Use and to approximate the value of each expression Use and to approximate the value of each expression Solve each equation. Check your solutions x = = x x = (x + 3) = (c + 3) (4c 1) = (x + 3) (2x 1) = 2

17 7-6 Common Logarithms Use a calculator to evaluate each expression to the nearest ten-thousandth. 1. log log log log log log Solve each equation or inequality. Round to the nearest ten-thousandth. 7. = = = = = = 15. = 16. = Express each logarithm in terms of common logarithms. Then approximate its value to the nearest ten-thousandth

18 7-7 Base e and Natural Logarithms Write an equivalent exponential or logarithmic equation. 1. = x 2. = ln 20 = x 4. ln x = 8 5. = ln (4x) = = 10x 8. ln = x Evaluate each logarithm to the nearest ten-thousandth. 9. ln 12, ln ln ln ln 943 ln ln 67 + ln ln 931 ln ln (139 45) Solve each equation or inequality. Round to the nearest ten-thousandth. 1. = = ln 6x 4 5. ln (x + 3) 5 = = ln (5x + 3) = = = ln (2x 5) = ln 5x + ln 3x > 9

19 7-8 Using Exponential and Logarithmic Functions 1. CARBON DATING Use the formula y =, where a is the initial amount of carbon 14, t is the number of years ago the animal lived, and y is the remaining amount after t years. a. How old is a fossil that has lost 95% of its Carbon-14? b. How old is a skeleton that has 95% of its Carbon-14 remaining? 2. Assume the population of gnats in a specific habitat follows the function P(t) =. a. Graph the function for t 0. b. What is the horizontal asymptote? c. What is the maximum population? d. When does the population reach 15,000?