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# Precalculus MATH 1330

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This 18 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1330 at University of Houston taught by Staff in Fall. Since its upload, it has received 430 views. For similar materials see /class/208417/math-1330-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

Math 1330 Section 54 Inverse Trigonometric Functions We are often interested in nding and working with the inverses of functions This will also be the case with trig functions But there s a problem For a function to have an inverse it must be onetoone and trig functions are not onetoone To get around this we begin by limiting the domain of each of the trig functions so that over the restricted domain the function is onetoone We ll work with the inverse functions for sine cosine and tangent in the streaming lectures The reciprocal functions are treated in the online text We ll start by looking at the inverse sine function Here are the graphs of the l functlons f x s1nx and gx E on the 1nterval 271 271 Th1s con rms that f x sinx is not a onetoone function 717 717 Ifwe 11m1t the doma1n of the functlon to 7 then the functlon 1s onetoone The range ofthis function is l l So on this limited interval we can define an inverse function We note the inverse sine function as So on this limited interval we can define an inverse function We note the inverse sine function as f x sin 1 x or f x arcsinx The domain of the inverse function will be 1 l and the range of the inverse function will be Here is the graph of fx sin 1 x Next we ll look at the inverse cosine function Since the cosine function is also not one toone we ll need to restrict the domain to an interval which will give us a onetoone function Then it makes sense to talk about an inverse function We can t use the same interval that we did for the inverse sine function however This interval does not produce a onetoone function Instead we ll use the interval 0 7 Here s the graph of f x cosx on the interval 0 It The range of this function is 1 1 So on this limited interval we can define an inverse function We note the inverse cosine function as fx cos 1 x or fx arccosx The domain of the inverse function Will be 1 l and the range of the inverse function will be 7r Here is the graph of fx cos 1 x Finally we ll look at the graph of the tangent function We ll restrict the domain to the 77quot 77quot interval 3 so that we can de ne an inverse function Here is the graph of fx tanx on the interval The range of this function is 0 00 J We can find an inverse tangent function on this limited domain We note the function as fx tan 1 x or fx arctanx The domain of the inverse function is 0 0 and 77quot 77quot the range is 7 Here is the graph of the inverse tangent function Most o en you will be expected to evaluate inverse trig functions That is given a problem such as sin391 x y you will want to nd the number in the interval 772 whose sine is x The simplest way to do this will be to rewrite the problem as sin y x and then use the chart we developed for working with unit circle values You will need to note the domain when working with problems ofthis type In some cases you can draw and label a right triangle in the appropriate quadrant or you can work with identities to evaluate These properties may be helpful when evaluating inverse trig problems sinsin39lx x on 71 l coscos39l x x on 71 l tantan39l 0 x on 7 cc 00 4 t 1 2 sm smx 7 x on 2 2 cosquot cosx x on 0 7 tanquot tanx x on Finally you will sometimes need to use a calculator Note that you will always use radians mode when evaluating inverse trig functions on your calculator Example 1 Evaluate cos39l Example 2 Evaluate arctan l Example 3 Evaluate sin 1T Example 4 Evaluate sec 1 2 Example 5 Evaluate csc 10 Example 6 Find the exact value sincos 1 Example 7 Find the exact value cos 1cos3 j Example 8 Find the exact value tansin 174 Example 9 Find the exact value cotsec 1 Example 10 Use a calculator to nd the exact value and round the answer to the nearest thousandth arcsin75 Example 11 Use a calculator to find the exact value and round the answer to the nearest thousandth cos 1 7 l 8 Example 12 Use a calculator to find the exact value and round the answer to the nearest thousandth sec 1 Example 13 Use a calculator to find the exact value and round the answer to the nearest thousandth cot 1 14 You can use graphing techniques learned in earlier lessons to graph transformations of the basic inverse trig functions Example 14 Sketch fx cos 1x 2 Exercise Set 11 An Introduction to Functions For each of the examples below determine whether the mapping makes sense within the context of the given situation and then state whether or not the mapping represents a function 1quot Erik conducts a science experiment and maps the temperature outside his kitchen Window at various times during the morning A Time Temp 39F 2 Dr Kim counts the number of people in attendance at various times during his lecture this afternoon 85 87 Time of People State whether or not each of the following mappings represents a function If a mapping is a function then identify its domain and range A B 4 2 A B 5 A B 6 8 quot A Math 1330 Precalculus The University of Houston Expre ss each of the following rules in function notation For example Subtract 3 then square would gt1 9 be written as fx x 32 Divide by 7 then add 4 Multiply by 2 then square Take the square root then subtract 6 10 Add 4 square then subtract 2 Find the domain of each of the following functions Then express your answer in interval notation 11 H N H Equot H 1quot H 5quot H P 5 x f H fx H xl goo quot4 x 79 3xl fx x24 2 fx 1 6x5 x illx28 got 23x15 x 8x720 ft J7 hemE good ht7 fxlx75 gx x7 J372x FOO x4 Cx 3 x77 Chapter 1 A Review QfFunctions Exercise Set 11 An Introduction to Functions 25 fx3 75 35 a fII b gt9II 26 gx3wx2x76 c hl9illl N l 277 78 h3 36 a fxx b 8xlx1 112 c hxx1 28 fx5 4x77 7 2 29 fakm 37 a fxx 4339 b gxx23i4 30 gt 751714 c hx2x235 Find the domain and range of each of the following functions Express answers in interval notation 3839 2 f0 I ll 6 31 a fxJ b gtltz67 b gx 76 c ht7 tz678 c hx I 7 6 d 1004 763 39 gl2x775 32 a ft3it 40 ht67171 b g0 374 c 05 41 fx35x674 d pa 37177 42 gx4873x2 33 a fxx274 b 8004 352 2 the domain and range of each of the following C h x 7 x 4 functions Express answers in interval notation d poo m Hint When nding the range rst solvefor x e 906 47362 43 a fx 2 x f rxlx24 x5 b goo x72 34 a ft 252 b gt 2 725 7 4 c ht 257 2 44 a x 7 f3 x d 111 IZSHZ b gx x7 3 e 11 42 7 25 t rl 42542 Math 1330 Precalculus The University of Houston Chapter 1 A Review QfFunctions Exercise Set 11 An Introduction to Functions 5 5 5 5 UI UI p UI N Evaluate the following 45 If fx5x74findi f6 A l n fa3 fa3 faf3 If fx3x1 find f6 8 le ft2 ft2 farlt2 If gx x2 73x 4 find go gH gem g9 gen ago If ht t2 2175 find ha he hc 6 7hx h2x 2hx 2x find x73 If x m7 e re r mks x2 If fx 7xfind2 x4 fa res ea31 fp32 2x75 if x24 If fx find 37x2 if x lt 4 f6 fa es f0 f4 re x24x if xlt72 find 772x if x272 If x res s f0 fa fH A7 Math 1330 Precalculus The University of Houston 3x2 if xlt0 53 If fx 4 if 0gxlt2find x5 if x22 o res A2 fa f4 re 74x77 if xgil 54 If x x26 if 1ltxlt3find 77 if x23 o fH fH f6 f6 14 Determine whether each of the following equations defines y as a function of x Do not graph 55 3x75y8 56 x3y2 57 x2y3 58 3x4 72xy5x 59 7x7y45 60 3x2 y2 16 61 x3y72y6 62 73y5 Chapter 1 A Review QfFunctions Exercise Set 11 An Introduction to Functions For each of the following problems 21 Find fxh b Find the difference quotient Assume that h at 0 67 x 7 7x 7 4 68 x 7 5 7 3x 69 x x2 75x72 70 x x2 7 3x8 71 x 7 78 72 x 6 Math 1330 Precalculus The University of Houston Chapter 1 A Review quunctions Math 1330 Section 32 Logarithmic Functions De nition For x gt 0 log x is the power to which b must be raised to get x where b gt0 b 72 1 We read this as log base b ofx We typically write logb x y and note that by x The form logb x y is called logarithmic form and the form by x is called exponential form Ifb 10 we write logx y and note that 10y x We call this the common logarithm If b e we write lnx y and note that ey x We call this the natural logarithm You should be able to write an equation that is in logarithmic form in exponential form and Vice versa Example 1 Write in exponential form lnx 3 Example 2 Write in logarithmic form 103x 4 You should be able to simplify expressions involving logarithms Example 3 Simplify log20125 Example 4 Simplify log5 125 Example 5 Simplify ln 6 Example 6 Simplify 81 Example 7 Simplify 6 You should be able to solve equations for x Example 8 Solve for x log2 64 x Example 9 Solve for x logx 9 l 2 Example 10 Solve for x loglx 5 1 Example 11 Solve for x log3 x2 4 2 Example 12 Solve for x log2log3x 2 Example 13 Solve forx 10 Example 14 Solve for x 65quot 4 Next we ll focus on graphing logarithmic functions Suppose f x logb x Where b gt 0 b 1 This is the logarithmic function with base I If 1 gt1 then the graph of this function has the shape shown here For this function domain is 0 0 and range is 0 0 The graph of the function passes through the key point 1 0 The graph has a vertical asymptote at x 0 If 0 lt b lt 1 then the graph of the function has the shape shown here For this function domain is 0 0 and range is 0 0 The graph of the function passes through the key point 1 0 The graph has a vertical asymptote at x 0 The functions f x bx and gx logb x are inverses of one another You should be able to graph log functions using transformations You should be sure to graph the key point and label the asymptote Example 15 Graph the function using transformations Label the key point and the asymptote fx10g3x 2 Example 16 Graph the function using transformations Label the key point and the asymptote fxlog5x2l Example 17 Graph the function using transformations Label the key point and the asymptote fx 10g4x 1 2 Example 18 Graph the function using transformations Label the key point and the asymptote f x 10g5x 3 Example 19 Graph the function using transformations Label the key point and the asymptote fx ln x 3 Example 20 Graph the function using transformations Label the key point and the asymptote fx 2 ln x 3 Finally you should be able to state the domain of a logarithmic function without graphing it Example 21 State the domain fx log2 x 3 Example 22 State the domain fx log5 3x l 4 Example 23 State the domain f x lnx2 5

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