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1 Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph represents a function.. Where is the function above discontinuous. Describe each tpe of discontinuit. 4. Using interval notation, describe the domain and range of the function above? 5. What are the domain issues ou must remember in this course? 6. The given function is onl drawn for > 0. Complete the function for < 0 with the following conditions: a) the function is ODD b) the function is EVEN 7. Suppose ou know the point (, ) is on the graph of a function. a) If the function is ODD, what other point is on the function? b) If the function is EVEN, what other point is on the function?

2 8. For each of the following functions, describe the domain in interval notation and identif an discontinuities (if the eist). Tr to do these without using our calculator. Then use our calculator to verif our analsis. 9 a) f 1 b) g Domain: Domain: Discontinuit: Discontinuit: c) h 4 d) k Domain: Domain: Discontinuit: Discontinuit: e) h f) p Domain: Domain: Discontinuit: Discontinuit: 9. Use the graph at the right to answer the following questions: ( 1.9, 4) a) Identif all etrema (.5, 0.5) b) Identif the intervals on which the function is increasing and decreasing. (1, 0). Use our graphing calculator to graph the function a) Identif all etrema. g. b) Identif the intervals on which the function is increasing and decreasing. 11. Complete the following questions from our tetbook: page 98: #1, 14, 8, 1,, 5, 7, 9, 40, 45

3 Precalculus Worksheet 1. Da 1. Complete the following questions from our tetbook: pages 99 0: #7, 79, 80, 81 In Calculus ou will need to read and write limit notation, so it makes sense to start using this notation now.. Rewrite the given end behavior for f () using limit notation. Sketch a possible graph for f (). a) As, f( ) b) As, f( ) c) As, f ( ) As, f ( ) As, f ( ) 0 As, f( ). Complete each limit notation for the end behavior of the function. A graph ma be helpful. a) f ( ) e lim f ( ) = b) f ( ) 1 lim f ( ) = - lim + 1 lim - 1 f ( ) = f ( ) = 4. For each function below, find the following: i) Domain ii) Vertical Asmptotes or Holes iii) Horizontal Asmptotes describe using limit notation a) f 1 5 b) g c) h 9 6 d) k 4

4 4 (continued). For each function below, find the following: i) Domain ii) Vertical Asmptotes or Holes iii) Horizontal Asmptotes describe using limit notation e) p f) q Eplain wh the function g( ) = does not have a horizontal asmptote When rational functions have no horizontal asmptote, ou must look for a slanted asmptote b dividing the numerator b the denominator (long division). When divided, the remainder will go to zero as goes to infinit, so the slanted asmptote is simpl the quotient of the division problem. The slanted asmptote is a function. Find the slanted asmptote of the following functions a) p( ) = b) g( ) = + 7. How do ou algebraicall prove a function is ODD or EVEN? 8. Complete the following questions from our tetbook:, page 99 #49 54 (prove algebraicall), 59, 6-66

5 Precalculus Worksheet Seven of the twelve basic functions have the propert that f (0) = 0. Which five do not?. Identif the four basic functions that are odd.. How man of the twelve basic functions are even? List them. 4. Identif the si basic functions that are increasing on their entire domains. 5. Identif the three basic functions that are decreasing on the interval (-,0). 6. Onl of the twelve basic functions are bounded. Which three? 7. Which of the twelve basic functions are not continuous? Identif the tpes of discontinuit in each function. 8. Identif the three basic functions with no zeros. 9. How man of the twelve basic functions have a range of all real numbers? List them.. Identif the four functions that do NOT have end behavior lim f ( ) =. 11. How man of the twelve basic functions have end behavior lim f ( ) =-? List them How man of the twelve basic functions look the same when flipped about the -ais? List them. 1. How man of the twelve basic functions look the same upside down as right-side up? List them. 14. How man of the twelve basic functions are bounded below? List them. 15. Complete #68 on page 11 in our tetbook. 16. Complete the following questions from our tetbook: page 9: #1 18, 5 4 (for #5 4, use a calculator to graph be sure to put our calculator in radian mode for #6 and #40.) 17. The graph of f ( ) = is one of the twelve basic functions. Guess which one, then graph f () on our calculator. Were ou right? If not, which of the basic twelve functions is f ()?

6 Piecewise Functions Piecewise Functions are simpl functions that have been broken into or more pieces where each piece is a portion of the graph with a limited domain. The limitations on the domain allow for the overall equation to pass the vertical line test, and thus be called a function. 18. The function = can be written as a piecewise function. Draw a graph of =, and then fill in the blanks below with the appropriate domain for each piece to complete the piecewise function representation of =. = ìï í ïî - if if 19. Write a piecewise function for the graph at the right. 0. Complete the following questions from our tetbook: page 1: #45 50 ask ourself self, do each of m graphs pass the vertical line test? 1. Your monthl cell phone bill is \$9.99 and includes 450 antime minutes. After that, ou pa \$0.45/minute. Write a piecewise function for the monthl cost of our cell phone as a function of the number of minutes ou talk.. In June 00, the cost, in cents, to mail a letter that weighs up to one ounce was \$0.7. The cost increased \$0. for each additional ounce (or portion thereof).

7 Precalculus Worksheet With a little practice, it is relativel simple to create a new function b adding, subtracting, multipling, or dividing two functions especiall if ou aren t asked to simplif the equations. When ou add, subtract, multipl or divide the functions in the problems from our tetbook, pa attention to the directions that sa state the domain. Find the domain of both functions, then find the domain of the function ou created. With that said Complete the following questions from our tetbook: page 17 #1, 5, 6 1. f ( ) = and g( ) = are shown in the graph below. Sketch the graph of the sum ( f g)( ) coordinates directl from the graph. Then graph the sum on our calculator and see how close ou came. + b adding the -. Net, ou are going to create composite functions. Again, think about what the domain of each function is separatel before finding the domain of the composite function. Complete the following questions from our tetbook: page 18 # You don t alwas have to create a new function in order to evaluate the function. If I write h( k ( 9) ) it means plug 9 into the function k, then plug the result into h. Complete the following questions from our tetbook: page 18 #9 and 5. In calculus, there is a rule called the chain rule (it is used to take a derivative of a composite function). In order to appl this rule, ou must determine what the inside and outside of a function is. This skill is equivalent to finding the two functions that make up a composite function. Complete the following questions from our tetbook: page 18 # 15, 17, and 0 6. If f and g are odd functions, show that the composite function f g is also odd.

8 7. The surface area S (in square meters) of a hot-air balloon is given b S( r) = 4pr, where r is the radius of the balloon (in meters). If the radius is increasing with time t (in seconds) according to the formula r() t = t, t > 0, find the surface area of the balloon as a function of the time t. 8. The volume V of a right circular clinder of height h and radius r is V = pr h. If the height is twice the radius, epress the volume V as a function of r The volume V of a right circular cone is V = pr h. If the height is twice the radius, epress the volume V as a function of r.. The surface area S (in square meters) of a hot-air balloon is given b S( r) = 4pr, where r is the radius of the balloon (in meters). If the radius is increasing with time t (in seconds) according to the formula r() t = t, t > 0, find the surface area of the balloon as a function of the time t The price p of a certain product and the quantit sold obe the demand equation p=- + 0, 0 < < Suppose that the cost C of producing units is C = Assuming that all items produced are sold, find the cost C as 5 a function of the price p. 1. Which ordered pair is in the inverse of the relation given b + 5= 9? A (, 1) B (, 1) C ( 1, ) D (, 1) E (1, ) 1. What does the Vertical Line Test tell ou about a graph? What does the Horizontal Line Test tell ou about a graph?

9 14. Complete the following questions from our tetbook: page 19 #9 4 Finding Inverses 15. Complete the following questions from our tetbook: page 19 #4, 45, A teacher gives a challenging algebra test. The lowest score is 5 which the teacher decides to scale to 70. The highest score is 88, which the teacher decides to scale to 97. a) Using the points (5, 70) and (88, 97) find a linear equation that can be used to convert raw scores (original) to scaled scores (new ). b) Find the inverse of the function defined b this linear equation. What does the inverse function do? Verifing (Proving) Inverses 17. Complete the following questions from our tetbook: page 19 #57, 59, To convert from degrees Celsius to degrees Fahrenheit, we use the formula f ( ) 5 degrees Fahrenheit to degrees Celsius, we use the formula g( ) 5 9 ( ) = = +. To convert from = = -. Show that f and g are inverses.

10 Precalculus Worksheet 1.5 For questions 1 8, do the following for each: a) Identif the parent function and describe the transformation b) Accuratel graph the function without using our calculator c) Identif the domain and range 1. 1 = ( ) = a) a) b) b) c) c). = 6-4. = ( +)² + 6 a) a) b) b) c) c)

11 5. - = - 6. = a) a) b) b) c) c) 4 7. = log ( + 5) 8. = - a) a) b) b) c) c)

12 9. Write an epression using f () that shows a reflection of f () across the a) the -ais and b) the -ais. a) b) For questions and 11, transform the given function b a) a vertical stretch b a factor of b) a horizontal shrink b a factor of 1/.. ( ) f = f ( ) = a) b) + a) b) For questions 1 and 1, suppose (6, ) is a point on the graph of = f () Which point must be on the graph of = f ( )? 1. Which point must be on the graph of ( ) A) (, 6) B) (6, ) A) (1, 4) B) (, 1) C) ( 6, ) D) (, 6) C) (1, 1) D) (, 4) f? For questions 14 and 15, use the graph of f () shown to the right to graph each transformation f ( + 1) f 1 For questions 16 and 17, use the graph of f () shown to the right to graph each transformation. 16. f ( - 6) = f ( + ) 4

13 18. Graph the following functions: a) h( ) = b) k( ) ( ) = log c) n( ) = d) ( ) p = - 1

14 Precalculus Worksheet Before ou can attempt word problems, ou must be able to translate from word phrases into algebraic epressions. Complete the following questions from our tetbook: page 15: #1 1 Another strateg is to define one variable in terms of another. This should give ou a function with variables.. If the length of a rectangle is 1 meter more than three times the width, then write an equation for the perimeter of the rectangle in terms of the width of the rectangle.. Complete the following questions from our tetbook: page 15: #15, 16, 17 Once the equation is set up, with enough additional information, ou can solve the problem. 4. The perimeter of a rectangle is 114 cm. The length is 6 cm less than twice the width. Find the dimensions of the rectangle. 5. Bob installed new windows in his house which were advertised to cut his energ bill b 7.1% per month after installation. If his new energ bill was \$15, what was his monthl bill before he installed the new windows? 6. Complete the following questions from our tetbook: page 15-15: # 1-, 6 You are now read to tackle the miture tpe problems. 7. Coffee worth \$.75 a pound was mied with coffee worth \$4.5 a pound to produce a blend worth \$4.11 a pound. How much of each kind of coffee was used to produce 40 pounds of the blended coffee?

15 8. A % solution of sulfuric acid was mied with an 18% solution of sulfuric acid to produce an 8% solution. How much % solution and how much 18% solution were used to produce 15 L of the 8% solution? 9. Reggie invests \$00, part at 7% annual interest and the rest at 8.5% annual interest. How much mone was invested at each rate if Reggie s total annual interest was \$900?. Complete the following questions from our tetbook: page 15: 8, 1, The final concept in modeling with functions is that one function can be used to determine the maimum or minimum for a certain variable. [Calculator Required] 11. Suppose a piece of cardboard 0 cm b 80 cm is used to make an open top bo b cutting out squares that are cm b cm from the corners. a) Epress the volume of the bo as a function of the length of the square that is cut out of each corner. b) What is the domain of this function? c) Find the maimum volume of the bo that can be made. What are the dimensions of this bo and what was the length of that was cut out of the original cardboard? 1. A 16 square meter rectangular pea patch is to be enclosed b a fence and divided into two equal parts b another fence parallel to one of the sides. a) What dimensions for the outer rectangle will require the smallest total length of fence? b) How much fence is needed?

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