Math 0 Handouts HW # (will be provided to class) Lines: Concepts from Previous Classes (emailed to the class) Parabola Plots # (will be provided in class) Functions Worksheet I (will be provided in class) Point Slope Equation of the Line 5 Parabola Plots # (will be provided in class) Functions Worksheet II 7 Introduction to chapters 4 and 5 9 Compound Inequalities Absolute Value Function Introduction to Set Notation 5 Functions Worksheet III 7 Factoring Polynomial Epressions 9 Cubic Function Flare Problem I 5 Functions Checkup 7 Which Function Reduces? 9 Rational Function Functions Worksheet IV Functions Worksheet V 5 Radical Review 7 Rational Eponents Practice/Powers of 9 Square Root Function 4 Domain/Range Practice I (supplemental) 4 Completing the Square Introduction 45 Complete the Square Presentation and Eplanations 47 Parabolas y a Changing the Shape 49 Functions Worksheet VI 5 Flare Problem II 5 Composition of Functions 55 Inverse Notes 57 Functions Worksheet VII 6 Graphs of Eponent Functions 6 Graphs of Logarithm Functions 65 Properties of Logarithms 67 Write Eponent/Logarithm functions 69 Compound Interest Practice 7 ~ ~
Circles and Non Linear Systems 7 Domain/Range Practice II 75 Final Eam Study Guide 77 Selected Solutions to the Final Eam Study Guide 79 HW Form A (4) 5 copies HW Form B (6) 5 copies HW Form C (8) copies HW Form D (graph) copies HW Form E () 8 copies Graph Paper 0 copies ~ ~
Math 0 Point Slope Equation of the Line. Write a Pt. Slope equation for each line described. The line with the given slope and passing through the given point. a) m,5 b) m 4 5 6, c) 7 m,8 The line passing through the given points., d),5 & e) 5, 5 &, f) 4, &, 5. List the given point and slope from each linear equation. a) y 5 b) y 4 c) y 5 m = m = m = (, ) (, ) (, ) 9. Write an equation of the line (pt. slope form) that is parallel to the given line and passes through the point (, 8). a) y b) y c) 47y 5 4. Write an equation of the line (pt. slope form) that is perpendicular to the given line and passes through the point, 4. a) y b) y c) 47y 5 ~ ~
5. Graph each line using the pt. slope equation. (Show and list the coordinates of the given pt.) a) y 5 b) y 4 c) y d) y 4 ~ 4 ~
Math 0 Functions Worksheet II Give the domain.... 4. 5. 6. 7. For #8 and 9, state the domain and range. Is it a function? 8. 9. {(4, 9), (, 7), (0, ), ( 5, 6), (4, 8)} D: D: R: R: Function? Function? ~ 5 ~
0. What are two reasons a number might need to get kicked out of the domain?... Consider the relation, 7,, 9,, 7 (a) Eplain why this relation is a function. (b) What ordered pair could you add that would make it fail to be a function?. For the function f, find each of the following: (a) f (b) f (c) f p (d) f a ~ 6 ~
Math 0 Introduction to Chapter 4: Previously learned material.. Is a solution to 5 8 7?. Is a solution to 5 8 7? Graph the solution to each inequality on the number line.. 4 4. 5. 5 6. 5 7. 4 8. 5 6 4 9. 47 ~ 7 ~
Math 0 Introduction to Chapter 5: Previously learned material. Simplify each epression. Use mental math. Do not show any work on this page.. 7 4 9. 5. 0. 7. 8. 4 4. 5. 54 5. 4 9. 7 6. 4 4 4. 5 7. 8 5. 4 8. Try these using mental math. 6. 4 7. 5 ~ 8 ~
Math 0 Compound Inequalities: Notes Algebraic Statement Number Line Interval Notation and 4 or 4 5 8 5 8, 6, 0, and 7 or 7 4 5 ~ 9 ~
Algebraic Statement Number Line Interval Notation 6 4 and 5 4,00, List the domain of the following functions using each of the requested notations. Algebraic/Word Statement Number Line Interval Notation y f 4 p g ~ 0 ~
Math 0 Absolute Value Function: f ( ) I. Review Write the function for each graph. Use proper notation.... II. Graph each Function. (Use the methods taught in class.) 4. f ( ) 5. f( ) 6. f( ) 7. f( ) ~ ~
8. f( ) 4 5 9. f( ) 0. f. g 4 III. Functions.. List the domain of each function. 5 a) f 4 D: b) g. Given p, answer the following. a) Evaluate p b) Evaluate p 4 5 D: c) Solve for all values of for which p 5 ~ ~
Math 0 Introduction to Set Notation Set Notation: a,5, q,,9 is read: The set containing a, 5, q,, and 9. A set is a collection of objects. The objects can be numbers, variables, symbols or anything else. The order in which the elements are listed does not matter. This is a discrete list of everything that is in the set. If you don t see an object/element listed, it is not included in the set. Set Builder Notation: Tells what an object/element of the set looks like is an even number between and 0 is read The set containing such that is an even number between and 0. Vertical line means such that Gives clear description how to decide whether or not an object is in the set This is a good notation for sets with infinite members or with lots of members. This is not a good notation for a set with only a few elements. Interval Notation: Interval notation looks like,, which is equivalent to. On the left side, you have the starting point, in this case, meaning that all negative numbers are included. On the right side, you have the ending point,. The parentheses are left indicates that is not included (we never include infinity since it impossible to actually arrive there). The bracket on the right indicates that the is included. This notation can only be used for infinite sets. The smaller value is always listed to the left and the larger value to the right. 9, means all real numbers starting at (and including) 9 and going up to (but not including). Equivalently, it means 9and, or 9. The set Set Notation Set Builder Notation Interval Not possible,,... is a natural number aeiou,,,, is a vowel, but not y Not possible 5 Not possible 5 Not possible Not possible,5,5 ~ ~
Write the set given on the number line in both set builder and interval notation:. 5. 6. / 4..8 5.7 5. What notation(s) is/are best for the following sets? (a) 0 (b) ~ 4 ~
Math 0 Functions Worksheet III Function Shape Shift from origin Coordinates of verte. (, ). (, ). f (, ) 4. (, ) 5. (, ) 6. (, ) 7. f (, ) 8. (, ) 9. (, ) 0. (, ). g6 8 (, ). (, ). (, ) 4. h 7 4 (, ) 5. (, ) 6. U 5 right, up ( 5, ) 7. V 6 left, down ( 6, ) 8. V up 4 ( 0, 4 ) 9. U left 8 ( 8, 0 ) ~ 5 ~
0. Write the function for each parabola: (a) (b) (c) (d) (e) (f) (g) (h) (i) ~ 6 ~
Math 0 Factoring Polynomial Epressions: sections 5. 5.7 Review and Notes The directions Factor each epression mean to write the epression as a multiplication problem. ) Common Factors: (This method should be tried first!) a) b) c) 4 8 4mb z b z y y 9 y 5 d) 5 p 5 e) by4 y 4 f) 4 ) Difference of Squares: a) 49 d) 4a 8b b) 44 e) 6 5 c) 5 m f) y m 4 ) Perfect Square Trinomials: a) 6 9 d) 4 8y 49y b) e) 9 0y 5y 4 c) 0y 5y f) 8 ~ 7 ~
4) Basic Trinomial epressions: a) 7 d) 4 b) 8 e) 9y 0y c) 40 f) p 5p 6 5) Trinomial epressions: a a) d) 6 5y y b) 5 e) 4 6 5 c) 0 y 4y 6) Sum/Difference of Cubes: y y y y y y y y a) 64 d) 5 b) c) 000 e) 8m f) 7 000y 6 a 8 ~ 8 ~
Factoring Polynomial Epressions (continued) 7) Miscellaneous: a) 5 45 b) 9 c) 4 6 d) 40 e) 8 f) 4 9 5 y g) 4 w h) i) b j) 5 4 7 5 4 0 k) 000 rr000 r ~ 9 ~
8) Here is a different way that factoring techniques might be used. These epressions do not have common factors. Fill in the right hand side of each statement so that they will be equivalent. a) m 7 b) 5y 4 5 c) 0 d) 5 4 4 e) 5 5 Additional Notes: ~ 0 ~
Math 0 The Cubic Function: y Graph each Function. (Use the methods taught in class.) ) y ) y ) y 4) y 4 5) y 6) y ~ ~
7) y 4 8) y 9) y 5 5 0) y Write the function for each graph. (Use proper notation.) ) ) ) ~ ~
Math 0 The Flare Problem I Eample : Position function: A flare is launched upwards from the ground with an initial velocity of 48 ft/sec. The height of the flare after t seconds is given by : ht 6t 48t. Find the height after second. after seconds. When did the flare hit the ground? Eample : Position function: A ball is dropped from the top of a 64 ft. tall building. The height of the ball after t h t 6t 64. Find the height after second. seconds is given by : When did the ball hit the ground? ~ ~
Eample : Position function: A flare is launched upwards from the top of a 8 ft. tall building with an initial velocity of ft/sec. The height of the flare after t seconds is given by : ht 6t t 8. Find the height after second. after seconds. When did the flare hit the ground? Eample 4: Position function: A flare is launched upwards from the top of a ft. tall building with an initial velocity of 96 ft/sec. The height of the flare after t seconds is given by : ht 6t 96t. Find the height after second. after seconds. When did the flare hit the ground? ~ 4 ~
Math 0 Functions Checkup Function definition Describe function Evaluate f a f subtracts one and then squares E: a a. f. f 5. f 5 f 4. f 5. 6. f Function definition Describe function Evaluate f p 4 f 7. f 8. f 9. f 5 0. f 4 + ~ 5 ~
. Give the domain in interval notation: (a) f (b) f (c) g 5 7 f (d) 6 5 (e) f (f) 4 f (g) g 5 (h) h 4 7 *(i) f *(j) f *=challenging ~ 6 ~
Math 0 Which Fractions Reduce? If a fraction does not reduce, circle it, otherwise reduce the fraction. ) ) ) 4) 5) 6) 7) 6 6 5 y 4 8 5 5 5 5 5 5 5 8) 5 5 9) 5 5 0) ) 5 5 5 05 ) 5 5 ) 54 ~ 7 ~
~ 8 ~
Math 0 The Rational Function: y Graph each Function. (Use the methods taught in class.) ) y ) y ) y 4) y ~ 9 ~
5) y 6) y 7) y 8) y 4 ~ 0 ~
Math 0 Function Worksheet IV Write each function. (Must be written in the form f() = or y = ) ) f() = ) ) 4) 5) 6) ~ ~
7) 8) 9) 0) Graph each function using the methods taught in class. (one square = one unit) ) y 5 ) y 4 ~ ~
Math 0 Functions Worksheet V Function Shape & up/dn Shift from origin Coordinates of verte (, ) (, ) (, ) (, ) (, ) (, ) Shift: Asymptotes: Shift: Asymptotes: Shift: Asymptotes: ~ ~
Shift: Asymptotes: Shift: Asymptotes: Shift: Asymptotes: ~ 4 ~
Math 0 Radical Review! (A review of the root symbol.) 4 5 00 9 6 4 49 69 64 6 44 8 0 9 5 4 0.6 4 7 0.0009 8 8 7 64 7 5 000 0 7 8 5 0.008 4 6 4 8 4 0,000 4 4 6 4 0.000 5 5 5 5 6 4 8 5 5 8 8 ~ 5 ~
What are the rules regarding the use of an absolute value in the evaluated radical? Key Terms: Inde: Radicand: Principal Square Root: Key eamples: even even odd Evaluate each radical epression: Use an absolute value symbol only when necessary. 4 = 4 = = 46 = m = 5 40 y = 8 40 y = 6 y = 5 = 5 = ~ 6 ~
Math 0: Rational Eponents Practice Rewrite each Eponent Epression into Radical form. Eponent Radical 5 4 7 5 y b 4 5 Rewrite each Radical Epression into Eponent form. Radical Eponent w 8 b 5 4m 5 4 m y ~ 7 ~
Math 0: Writing Epressions as Powers of. Write each of the following epressions in the form: p where p is a real number. Eamples: 5 6 ) 7 ) 4 ) 7 7 4) 7 5 5) 5 4 6) 5 7) 4 8) 5 4 9) 0) 4 ) 5 ) 5 ) 4 4) 4 5) 4 6) 7) 8) ~ 8 ~
Math 0 Square Root Function: y Graph each Function. (Use the methods taught in class.) ) y ) y ) y 4) y 5) y 6) y 7) ~ 9 ~
y 4 8) y 9) y 5 0) y (plot points) REVIEW: y 4 ) ) y ~ 40 ~
Math 0: Domain and Range Practice I ) Circle all functions with a Domain of all Real Numbers. y f g y y 5y ) List the Domain of each function. (Use interval notation.) g y y f 5 ) Circle all functions with a Range of all Real Numbers. y f g y y 5y 4) List the Domain and Range for each graph. (Use interval notation.) D: D: R: R: ~ 4 ~
Problems 5 6: Use the list of functions to respond to each question. 5 4 g r p f 5) Evaluate each of the following. Simplify if possible. g f p 4 f g f g p a g a 6) Solve each of the following for all values of the given variable such that: a. f 4 b. p a 7 ~ 4 ~
Math 0 Completing the Square introduction. Simplify each epression. a) 5 = c) 4 = b) = d) =. Factor each epression and write as a perfect square of a quantity. a) b) 4 4 d) 8 6 e) 6 9 4 c). Write the number in the blank spot that completes the square. Factor the epression and write as a perfect square of a quantity. a) b) 4 c) 0 d) 6 5 4. Solve each quadratic equation by using the square root rule. a) 49 d) b) 5 e) 00 56 c) 8 8 f) 4 0 ~ 4 ~
5. Solve each quadratic equation by using the completing the square method. a) 8 9 d) 4 0 b) 4 5 e) 80 0 c) f) 74 0 6) Solve each quadratic equation by using the completing the square method. a) b) 4 4 0 c) 8 7 0 5 0 ~ 44 ~
Math 0 Completing the Square: Presentation and Eplanation Eample. Solve the equation using Complete the Square. 45 0 (This does factor so we could check using the zero product property at the end.) 4 5 Addition property of equality. (Added 5 to both sides of the equation.) 449 5 49 Addition property of equality. (Add 49 to both sides of the equation.) 7 64 Factored the left hand side and simplified the right hand side of the equation. 7 64 Square root rule. (This method generates two possibilities.) 7 8 or 7 8 These are the two equations that must be solved. or 5 The solutions.. Solve the quadratic equation by using the completing the square method. Follow the same steps as done in the eample above. Fill in any missing information in the eplanation. 80 0 (This does factor so we could check using the zero product property at the end.) Addition property of equality. (Added to both sides of the equation.) Addition property of equality. (Add to both sides of the equation.) Factored the left hand side and simplified the right hand side of the equation. Square root rule. (This method generates two possibilities.) These are the two equations that must be solved. The solutions. ~ 45 ~
. Circle all the errors committed in solving each of these quadratic equations. Eplain the nature of the errors. (Note: Errors might also be in the notation.) Rework only the steps necessary to the right and give the correct solution. a. c. 45 0 45 8 6 45 6 9 i 5i or 9i b. d. 07 0 69 0 0 7 6 9 000 7 699 9 0 7 8 0 i 7 8 0 i 7 or 5 ~ 46 ~
Math 0 Parabolas: Changing the Shape Graph each Function. (Use the methods taught in class.) ) y and y ) y ) y 4) y y 4 6) 5) y ~ 47 ~
~ 48 ~
Math 0 Functions Worksheet VI Equation verte min/ma range AoS Tall/Short/Same (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) y 4 * (, ) (, ) (, ) ~ 49 ~
Using all the information you found, graph the equations with *. Find two points on either side of the AoS. Choose wisely. 4.. y ~ 50 ~
Math 0 The Flare Problem II ) Suppose that a flare is launched upward from the ground with an initial velocity of 64 ft/sec. Its height in feet, h(t) after t seconds is given by: ht ( ) 6t 64t. Answer the following questions. Show your work and use the correct unit of measurement. What is the height of the flare after second? When will the flare hit the ground? When will the flare reach its maimum height? How high will the flare go? ) Suppose that a ball is dropped from the top of a 40 foot tall building. Its height in feet, h(t) after t seconds is given by: ht () 6t 40. Answer the following questions. Show your work and use the correct unit of measurement. What is the height of the flare after second? When will the flare hit the ground? When will the flare reach its maimum height? How high will the flare go? ~ 5 ~
) Suppose that a flare is launched upward with an initial velocity of ft/sec from a height of 48 ft. Its height in feet, h(t) after t seconds is given by: ht ( ) 6t t 48. Answer the following questions. Show your work and use the correct unit of measurement. What is the height of the flare after second? When will the flare hit the ground? When will the flare reach its maimum height? How high will the flare go? 4) Suppose that a flare is launched upward with an initial velocity of 40 ft/sec from a height of 70 ft. Its height in feet, h(t) after t seconds is given by: ht ( ) 6t 40t 70. Answer the following questions. Show your work and use the correct unit of measurement. When will the flare reach its maimum height? How high will the flare go? ~ 5 ~
Math 0 Composition of Functions Reminder: A function is. DEFINITION: The composite function f g, the composition of f and g, is written as f g This is pronounced f of g of and can also be written f g. Eample : Let f ( ) g ( ) Evaluate a. f g4 c. f g b. g f 4 d. g f f( ) Eample : Let g ( ) Evaluate a. g( f ()) c. g( f( )) b. g f 4 d. f g ~ 5 ~
Eample : Let f ( ) g ( ) Evaluate a. g( f (5)) c. f ( g ( )) b. f g5 4. g f Eample 4: A Simple use of composition of functions. Eample: A car salesperson s monthly salary is $,000 base salary plus $50 for each car that they sell. Let n be the number of cars and S(n) be the salary function. Epress the salary function in terms of number of cars. S(n)= Each month, his employer contributes $00 plus 5% of the salesperson s salary to their retirement. If R represents the amount put into retirement, then R (in dollars) is a function of S. R(S)= Now, R is a function of S, and S is a function of n. Can we epress R as a function of n? Meaning if we want to find his retirement payment based on the number of cars they sell per month.. Interpretation from the salary eample: R Sn R Sn R R S n ~ 54 ~
Math 0 Inverses of functions: NOTES. List the 4 Biggies of Inverses. I. II. III. IV.. Basic Eamples. f, Given that 4,7,,8, 6,0 f. Graph the function and its inverse on the same ais. Graph the line y = on each ais. Is the inverse a function? (a) f ( ) 5 (b) g ( ) 4 Is the inverse a function? Yes No Is the inverse a function? Yes No ~ 55 ~
p ( ) (d) A ( ). (c) Is the inverse a function? Yes No Is the inverse a function? Yes No (e) p ( ) (f) Graph of function is given. Is the inverse a function? Yes No Is the inverse a function? Yes No ~ 56 ~
4. Find the Inverse of each function. 5. Verify that the functions are inverses using composition of functions. (a) f ( ) 5 (a) Goal: Show that f ( ) Work: g ( ) 4 (b) (b) g ( ) (c) B ( ) 4 (c) B ( ) (d) q ( ) 4 7 (d) q ( ) ~ 57 ~
4. Find the Inverse of each function. 5. Verify that the functions are inverses using composition of functions. (e) p ( ) 6 (e) p ( ) (f) 4 r ( ) 5 (f) r ( ) (g) h ( ) 5 (g) * a fun challenge h ( ) ~ 58 ~
Math 0 Functions Worksheet VII Write each function. (Must be written using proper notation.) ) f() = ) ) 4) 5) 6) 7) 8) 9) ~ 59 ~
0) ) ) (problems 4) Graph each function using the methods taught in class. (one square = one unit) ) y y 4 4) 5) Convert the function into verte form. List the coordinates of the verte. State the domain and range of the function. y 4 6) Write the verte form of the parabola with verte at, that also passes through the point 5,. ~ 60 ~
Math 0 Graphs of Eponent Functions Graph each eponent function.. y. y. y 4. y 5. y 5 6. y ~ 6 ~
7. y 8. y 9. y 0. y. y. y ~ 6 ~
Math 0 Graphs of Logarithm Functions. Graph y and y log (Draw the line y = also.). Graph y log Eponent Log D: D: R: R: ( 6) Describe the transformation to each graph. List the Domain and Range for each function. y log 5. y log D: D: R: R: 4. y log 6. y log D: D: R: R: ~ 6 ~
Sketch each function. Draw the asymptote as a dotted line. 7. y log 8. y log 9. y log ~ 64 ~
Math 0 Properties of Logarithms Identify whether each statement is True or False (T or F). If the statement is false, identify the error. ) log 5log log 5 4 4 4 ) log log5 log7 ) log log5 log7 4) log mlog n log0 mn 5) log y log y 6) log 4 w 4log w log 7) log log y log y 8) log log y log y 9) log 5 log 5 log 7 0) log 000.04 7log 000.04 ~ 65 ~
Solve each equation. ) log 7 log ) 4 4 log 7 4 ) log log 6 4) log log 9 log 0 7 7 7 ~ 66 ~
Writing Eponent Functions: Write each eponent function (base ). Use correct notation. Write the Domain and Range for each using interval notation. ) 4) ) 5) ) 6) ~ 67 ~
Writing Logarithm Functions: Write each Logarithm function (base ). Use correct notation. Write the Domain and Range for each using interval notation. ) 4) ) 5) ) 6) ~ 68 ~
Math 0 Compound Interest Practice: ) Set up a compound interest epression for each set of given values. (Do not evaluate.) a) How much money will we have in years if we invest $4,000 in an account earning.5% compounded annually? b) How much money will we have in 9 years if we invest $6,700 in an account earning 4% compounded quarterly? c) How much money will we have in 8 years if we invest $,00 in an account earning 5.% compounded continuously? ) Set up and solve each compound interest problem. (Eact solutions only.) a) How much money do we need to deposit in order to have $5,000 at the end of 6 years in an account earning.4 % compounded continuously? b) How much money do we need to deposit in order to have $,000 at the end of years in an account earning.9 % compounded monthly? c) What rate of interest is required to double our money in 0 years in an account earning interest on an annual basis? d) What rate of interest is required to double our money in 0 years in an account earning interest compounded quarterly? ~ 69 ~
) Set up a compound interest epression for each set of given values. Solve for t using common or natural logarithms. (Show your work.) a) How long will it take for $,500 to increase to $4,000 in an account earning.% compounded annually? b) How long will it take for $500 to increase to $600 in an account earning 4.7% compounded monthly? c) How long will it take to double our money in an account earning 4.% compounded continuously? d) How long will it take to triple our money in an account earning.8% compounded continuously? ~ 70 ~
Math 0 Circles and Non Linear Systems List the center and radius of each circle... y 6 center Radius: r = y 0 center Radius: r =. y 4 center Radius: r = 4. y 5. 6. y 4 5 center Radius: r = y 7 center Radius: r = 9 center Radius: r = 7. y y 0 5 6 9 0 center Radius: r = 8. y y 8 0 center Radius: r = Graph each circle. 9. y 6 0. y y 4 0. y 6 0 ~ 7 ~
Solve the non linear systems by graphing. y. y. y y Solve each non linear system. y 4. y 5. y 9 y4 5 6. y y 5 ~ 7 ~
Math 0: Domain and Range Practice II ) Circle all functions with a Domain of all Real Numbers. y y g f y 4y 7 ) List the Domain of each function. (Use interval notation.) y 4 g 4 y ln y 4 f 4 y log ) Circle all functions with a Range of all Real Numbers. y y g f y 4y 7 4) List the Domain and Range for each graph. (Use interval notation.) D: D: R: R: ~ 7 ~
5) List the range of each function. (Use interval notation.) f 4 y p 8 6) Write a radical function with domain,. 7) Write a rational function with domain,,. 8) Write an eponential function with range,. 9) Write a logarithmic function with domain,. 0) List the domain of each function. (Use interval notation.) f 56 f f 9 f 4 ~ 74 ~
Math 0 Final Eam Study Guide The Final Eam is comprehensive but identifying the topics covered by it should be simple. Use the previous eams as your primary reviewing tool! This document is to help provide you with more specific eamples related to functions and function notation. These concepts are thoroughly covered by the tetbook but are not isolated to any particular section or chapter. Every chapter (ecept chapter ) contains problems related to evaluating functions at particular values and solving equations/inequalities involving function notation.. Know the difference between an equation, an epression and a function. Know which instruction might be associated with each, i.e., simplify, solve and graph.. Be able to graph any of the following base functions with left/right and/or up/down shifts. a. Linear b. Absolute value c. Cubic d. Radical e. Rational f. Quadratic g. Eponential h. Log. Be able to recognize the difference between rational, irrational, and imaginary numbers. Refer to Eam 4. 4. Functions and their inverses. The 4 Biggies. Handout with solutions is available on the website. ~ 75 ~
Use the following functions to respond to questions 5 through. Rationalize all denominators. Use i notation with imaginary solutions. f( ) 5 g ( ) ( ) p ( ) ( ) log k h( ) r ( ) 5 q( ) w ( ) 6 n ( ) ( ) j ( ) log ( ) 5. Use interval notation to identify the domain and range of any of the functions listed above (ecept n() and j(). 6. Evaluate and simplify. f () 7. Find all values of for which a) f ( ) p( ) b) r ( ) c) g ( ) d) w( ) p( ) e) w ( ) 0 f) h ( ) n ( ) g) k ( ) h) k ( ) j ( ) 8. Find k ( ) 9. Find p( f( )) 0. Rewrite w() in verte form. w( ). Use long division or synthetic division to perform. p( ) wb ( ) w(). Evaluate and simplify. b ~ 76 ~
Math 0 Selected Solutions to the Final Eam Study Guide. Know the difference between an equation, an epression and a function. Know which instruction might be associated with each, i.e., simplify, solve and graph.. Be able to graph any of the following base functions with left/right and/or up/down shifts. i. Linear e) Rational j. Absolute value f) Quadratic k. Cubic g) Eponential l. Radical h) Logarithm. Be able to recognize the difference between rational, irrational, and imaginary numbers. Refer to Eam 4. Eamples of: Rational numbers: Irrational numbers: 7 6,, 5,0,, 8, e,, 5, 5 Imaginary numbers:, i 4, 8 4. Functions and their inverses. The 4 Biggies. Handout with solutions is available on the website. ~ 77 ~
Use the following functions to respond to questions 5 through. Rationalize all denominators. Use i notation with imaginary solutions. f( ) 5 g ( ) ( ) p ( ) ( ) log k h( ) r ( ) 5 q( ) w ( ) 6 n ( ) ( ) j ( ) log ( ) 5. Use interval notation to identify the domain and range of any of the function listed above. g ( ) ( ) Domain: All reals., Range: Since the verte is at, and the parabola opens up, the range is all y values greater than or equal to., f( ) 5 Domain: The value in the square root must not be negative. 5 0 5 5, h ( ) Domain: All reals., Range: The y values are all positives. 0, k ( ) log This is the inverse of h(). Domain and range are switched. Domain: 0, Range:, 6. Evaluate f () 5 7 7 7 7 7 7 ~ 78 ~
7. Find all values of for which a) f ( ) p( ) 5 Square both sides 5 Now it is quadratic. Zero product property. Check your solutions. 0 4 b) r ( ) 5 Cube both sides c) g ( ) ( ) Add one to both sides. ( ) 0 Square root rule. (two solutions) d) w( ) p( ) 6 set = 0 5 0 Doesn t factor. Use another method. e) w ( ) 0 6 0 Doesn t factor. Try completing the square. f) h ( ) n ( ) () Same bases, therefore eponents are equal. Now solve for. g) ( ) k h) k ( ) j ( ) log Translate into an eponent equation. 9 log log ( ) Same bases. Arguments are equal. Now solve for. ~ 79 ~
8. Find k ( ) To find the inverse, switch and y and then solve for y =. You will need to translate from logarithm to eponent to succeed. p f p( f( )) p 5 9. Find ( ( )) p or. 0. Rewrite w() in verte form. Change to the form of the function that indicates where the verte is. w ( ) 5 Solution: w( ). Use long division or synthetic division to perform. p( ) Note: p( ) is Then w p ( ) ( ) 6 Long division: 6 Synthetic division: 6. Evaluate and simplify. wb ( ) w() b wb ( ) w() b 6 6 b b b = 4 4 46 4 b b b b b 6b4 4 b b 6b b 6 b ~ 80 ~