Part 1: No Calculator Evaluating Functions and Determining their Domain and Range The domain of a function is the set of all possible inputs, which are typically x-values. The range of a function is the set of all possible outputs, which are typically y-values. When determining the domain of a function based on the formula: Exclude any numbers that cause division by zero. Exclude any numbers that cause the square root of a negative number to occur. 1. Algebraically determine the domain of the following functions. State each domain using set-builder notation AND interval notation. Then find f(0) and f( 2) for each function. Function Domain (SB Notation) Domain (Int. Notation) f(0) f( 2) f(x) = 5x + 7 f(x) = 2x + 3 f(x) = 3(x 1) 2 f(x) = 3x 6 f(x) = x+1 3x 6 f(x) = x 6 x 2 +x 6 2. State the domain and range of each function below using interval notation. a) Domain: Domain: Range: Range: 6 Figure 1 y Figure 2 y 6 2 x 2 x 2 2 6 8 2 6 2 2 6 2 6 6 1
Evaluating, Solving, and Determining Domain and Range Using the Graph of a Function 3. Use the graph of y = h(x) in Figure 3 to answer the following. State all solutions to equations in a solution set. y Figure 3 2 x 2 2 6 8 10 12 1 2 a) State the domain of h. State the range of h. c) Evaluate h(0). d) Solve h(x) = 0. e) State the following: f) Evaluate h(1). x-intercept(s): y-intercept: g) Solve h(x) = 1. h) Solve h(x) = 3. i) Solve h(x) 2. j) Solve h(x) > 3. Part 1: No Calculator Page 2
Sketching Graphs of Functions By Hand. Sketch the graph of y = f(x) for each of the following functions WITHOUT a calculator. Clearly label any x-intercepts, y-intercepts, and vertices. a) f(x) = 1 3 x 1 Figure c) f(x) = x + 3 Figure 6 f(x) = (x 1) 2 + Figure 5 d) f(x) = x Figure 7 Part 1: No Calculator Page 3
5. Let p(x) = x 2 9x + 7. Math 95 Practice Final Exam Simplifying Expressions with Function Notation a) p(x 3) p( 5x) c) p(x) + d) 8p(x) Completing the Square 6. Write the quadratic function q(x) = x 2 + 12x + 5 in vertex form by completing the square. Identify the vertex. 7. Solve quadratic equation x 2 6x + 1 = 10 using completing the square. Clearly state all real and complex solutions in a solution set. Part 1: No Calculator Page
Solving Equations and Inequalities Algebraically 8. Algebraically solve the following equations and inequalities. Show all your work. a) 8 < x + 12 16 2x 9 = 7 c) x + 6 10 d) 2x + 1 > 9 Part 1: No Calculator Page 5
9. Algebraically solve the following equations. Show all your work and include your check(s). a) 2x + 18 + 3 = x 8 x 2 = x x + 2 c) 5x 2 x = 1 (State all real and complex solutions) Part 1: No Calculator Page 6
Rational Exponents and Radical Notation 10. Simplify each expression below as much as possible. a) 25 1/2 100 1/2 c) 36 1/2 d) 6 1/3 e) ( 8) 2/3 f) 16 3/ 11. Write each expression below using radicals. a) (2x) 3/7 2x 3/7 c) y /5 12. Simplify each expression below using the rules of exponents. Write all simplified answers using positive, rational exponents. Assume all variables are positive. a) (x 6 ) 1/2 y 1/2 c) 5x 1/2 x 2/3 y 1/8 d) (8v 12 ) 1/3 e) ( s 10 t 6 ) 1/2 f) ( 2p 3/5 p /5) 0 Part 1: No Calculator Page 7
Simplifying Rational Expressions There are four important facts about rational expressions and equations: Only factors cancel. Restrictions occur when a factor is cancelled from the numerator and denominator and need to be stated. A rational expression is equivalent to another rational expression when BOTH the numerator and denominator are multiplied by the SAME expression. An equation with rational expressions is equivalent to another equation when BOTH sides of the equation are multiplied by the SAME expression. 13. Perform the indicated operation and simplify the expression as much as possible. Include any necessary restrictions. t 2 + 9t + 8 t + 8 a) 2t 6 2x + x 2 5x + 6 x2 + x + x 8 Part 1: No Calculator Page 8
1. Perform the indicated operation and simplify the expression as much as possible. Include any necessary restrictions. y + 2 a) y 3 y + 1 y + 3 x + 9 + 15 x 2 + 13x + 36 Part 1: No Calculator Page 9
15. Simplify the complex fractions below as much as possible. Include any necessary restrictions. 3 a) 3 x x 3 + 2 t t 2 16 t 16 t 2 + t Part 1: No Calculator Page 10
Math 95 Final Exam Part 2: Calculator Permitted Applications 16. The height of a golfball, h(t) (in meters), t seconds after the golfball is hit is be modeled by: h(t) =.9t 2 + 18t + 3 a) Find h(2.5) and explain what this means in the context of this model. Solve h(t) = 15 and explain what this means in the context of this model. c) What is the maximum height of the golfball? When does this occur? Part 2: Calculator Permitted Page 11
17. When an object is dropped, the time it takes to hit the ground, t, in seconds, can be modeled d by t = where d stands for the initial height of the object in feet. Algebraically determine the 16 following. a) If an object is dropped from the top of a 150-foot-tall building, how long will it take for the object to hit the ground? If an object fell for 2 seconds before hitting the ground, from what height was it dropped? 18. A drug is administered, and the amount of the drug in a person s bloodstream (in mg) after t hours is given by A(t) = 10t t 2 +. a) Find and interpret A(1.5). Solve and interpret A(t) = 2. Part 2: Calculator Permitted Page 12
Answers 1. Function Domain (SB Notation) Domain (Int. Notation) f(0) f( 2) f(x) = 5x + 7 {x x is a real number} (, ) 7 17 f(x) = { } ( 2x + 3 x x 3 2, 3 2] 3 7 f(x) = 3(x 1) 2 {x x is a real number} (, ) 7 31 f(x) = 3x 6 {x x is a real number} (, ) 6 12 f(x) = x+1 3x 6 {x x 2} (, 2) (2, ) 1 6 f(x) = x 6 x 2 +x 6 {x x 3, x 2} (, 3) ( 3, 2) (2, ) 1 2 2. a) Domain: (, ) Domain: [ 6, ) ( 1, 1] Range: [, ) Range: ( 3, 1] (2, 6] 7 12 3. a) (, ) (, 3] c) 2 d) x = 2, x = 8; {2, 8} e) x-intercept(s): (2, 0) and (8, 0) f) 1 y-intercept: (0, 2) g) x = 3, x = 7; {3, 7} h) x = 5; {5} i) {x x or x 6} j) {x 1 < x < 11} (, ] [6, ) ( 1, 11). a) f(x) = 1 3 x 1 f(x) = (x 1) 2 + c) f(x) = x + 3 d) f(x) = x 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 5. a) x 2 33x + 70 100x 2 + 5x + 7 c) x 2 9x + 11 d) 32x 2 72x + 56 6. q(x) = (x + 6) 2 31; the vertex is ( 6, 31) 7. {3 i 2, 3 + i 2} 8. a) {x 1 x < 5} { 1, 10} c) {x 16 x } d) {x x < 5 or x > } [ 1, 5) [ 16, ] (, 5) (, ) 9. a) {9} {} c) { 2 5 1 5 i, 2 5 + 1 5 i} 1 10. a) 5 c) 6 d) e) f) 8 10 Answers Page 13
7 11. a) 8x 3 2 7 x 3 1 c) 12. a) y 3/8 c) 5x 7/6 1 x 3 d) 2v s e) 5 f) 1 t 3 13. 2 8, for t 8 (t 3)(t + 1) (x 3)(x + 2), for x 2 1. a) 8y + 11 3 (y + )(y 3), for x 9 x + 15. 1 x 2, for x 3 t 2, for t, t 0 (t ) 2 16. a) h(2.5) = 17.375. After 2.5 seconds, the height of the golfball is 17.375 meters. t 0.875, t 2.798. The height of the golfball is 15 meters after about 0.875 seconds and again after about 2.798 seconds. 5 y c) The maximum height is about 19.531 meters and occurs after about 1.837 seconds. 17. t = 5 6 a) 3.062 It will take about 3.062 seconds. d = 6. The height was 6 feet. 18. a) A(1.5) = 2.. After 1.5 hours, there are about 2. milligrams of the drug in the patient s bloodstream. t = 1 or t =. There are 2 milligrams of the drug in the bloodstream after 1 hour and again after hours. Answers Page 1