ࡱ> MO LB5@ 8bjbj22 -XXc/6666b!b!b!b!F"L|5"****9,9,9,4444444$26R8T!59,,"9,9,9,!566**%653339,68*:*439,4334n4*" :),b!Q-x44L50|5481846666849,9,39,9,9,9,9,!5!5^3^G. Exponents and Exponential Functions 1. Exponential Functions Def/ An exponential function is defined by: y = ax where  EMBED Equation.DSMT4  D = Reals R =  EMBED Equation.DSMT4  What problems occur: (a) if a < 0 (negative)? Well, let a = -4. Then y = (-4)x might look exponential but if x = 1/2, we would have troubles with numbers like  EMBED Equation.DSMT4 ? (b) if a = 0? Well, y = 0x would just be a horizontal line (constant function) y = 0 which is not considered an exponential function. Also, there would be a hole in the graph at (0,0) since 00 is not defined. (c) if a = 1? Well, consider the function y = 1x. This would just be the horizontal line or constant function, y = 1. Well-defined but we wont consider this an exponential function either. Thm/ y = ax is an increasing function iff a > 1 Thm/ y = ax is a decreasing function iff 0 < a < 1  y = 3x (incr) vs y = (1/3)x (decr)  More notes: (i) All exponential functions contain the point (0,1). (ii) y = 3x and y = (1/3)x (taken together) are symmetric wrt the y-axis. (iii) y = 0 is the horizontal asymptote. (iv) y = 3-x is equivalent to y = (1/3)x. Power functions also involve exponents but each power function has a constant power (exponent) and a variable base (y = xn) while an exponential function has a constant base and a variable exponent (y = ax). Exponential functions employ a functional notation that is looks exponential. As a exercise, consider the possible function notation:  EMBED Equation.DSMT4 = f(x). Then (a) f(4) = exp3(4) = 34 or 81 (b) 2 EMBED Equation.DSMT4 exp3(4) = 162 (c) f(x+2) = exp3(x+2) = 3x+2 = 3x EMBED Equation.DSMT4 32 = exp3x + exp32 = f(x) + f(2) Of course, f(a+b) does not in general equal f(a) + f(b). G. Exponents and Exponential Functions (continued) 2. Exponents Laws of Exponents (work well if the base a is positive!)  EMBED Equation.DSMT4  *All of the above statements should be qualified since we cannot divide by zero. Although exponential functions must have positive bases (a > 0, a  EMBED Equation.DSMT4 ), exponential expressions may have negative bases as in: (-2)3 = -8. Also, although the above laws hold for fractional and irrational exponents, the expressions must be well-defined. Usually the fine print requires that the base, a, be positive. Ex/  EMBED Equation.DSMT4 Incorrect if  EMBED Equation.DSMT4  since  EMBED Equation.DSMT4 . Integer exponents have been used since the first year of algebra. Rational exponents can be troublesome as well soon see. Lets go back to our work with radicals to see what kind of problems we will run into. Ex/  EMBED Equation.DSMT4  Yes? No? Well Lets try  EMBED Equation.DSMT4 ? Wrong answer! Remember complex (imaginary) numbers?  EMBED Equation.DSMT4  Ex/  EMBED Equation.DSMT4  (See Law (6) above) Yes? No? Well not if a = -4 and b = -9. Ex/  EMBED Equation.DSMT4  Yes? No? Well What if a = -4?  EMBED Equation.DSMT4  Wed better be careful about reducing fractions when the bases are negative! G. Exponents and Exponential Functions (continued) 3. Fractional Exponents and Radicals First of all, think  EMBED Equation.DSMT4 . Now lets consider rational exponents. Def/  EMBED Equation.DSMT4  For  EMBED Equation.DSMT4  (positive bases), we also have  EMBED Equation.DSMT4 . So whats the problem? Really none so long as the base is not negative. Heres the final word on negative bases. Dont bother with this on a first reading. Def/  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4  (negative base!) and  EMBED Equation.DSMT4  IF  EMBED Equation.DSMT4  IF NOT, all bets are off! Ex/ (Just for fun!) Lets see what trouble we can get into! Try #1:  EMBED Equation.DSMT4  Try #2:  EMBED Equation.DSMT4  Try #3:  EMBED Equation.DSMT4  (TI-83 agrees!) *Important Ex/ So all we have to do is reduce our fractions? Right? Wrong! Consider  EMBED Equation.DSMT4  Now if all we have to do is to reduce 4/4 to 1, then we get  EMBED Equation.DSMT4 . We (incorrectly) set  EMBED Equation.DSMT4 . Our (incorrect) conclusion would be:  EMBED Equation.DSMT4  The actual answer is of course  EMBED Equation.DSMT4  Memorize this:  EMBED Equation.DSMT4  Youll be happy to hear the final word The power of a negative base is UNDEFINED for irrational exponents. Try these undefined expressions on your calculator:  EMBED Equation.DSMT4 = (-8)^ EMBED Equation.DSMT4 = ? or  EMBED Equation.DSMT4  Lets stick with positive bases and continue! Our calculator will take care of irrational exponents. Negative exponents are easy. Negative rational exponents are handled just like negative integer exponents. Def/  EMBED Equation.DSMT4  (Take your pick!) for a > 0, etc. Restrictions on variables are not always easy to spot. Ex/ x-2 = 1/x2  EMBED Equation.DSMT4  Ex/  EMBED Equation.DSMT4  G. Exponents and Exponential Functions 3. Fractional Exponents and Radicals (continued) Thm/ For  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  Since  EMBED Equation.DSMT4 . 4. Equation Solving Solving irrational equations involves the unknown quantity (usually denoted by x) under a radical sign (in the radicand). Ex/ (Try to follow the logic of the argument carefully.) (A)  EMBED Equation.DSMT4  Well move the x over and square both sides. Right?  EMBED Equation.DSMT4  Notice the double implication or equivalence here. (B)  EMBED Equation.DSMT4  Now here we have only the single arrow  (back to double arrows)  EMBED Equation.DSMT4  (now some cool factoring)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  These last four equivalence symbols indicate that the truth sets for these last 5 equations are the same: TB =  EMBED Equation.DSMT4  but this is not the truth set for our original equation. We do know that any solution for A is also a solution for B (TA  EMBED Equation.DSMT4 TB). So we take all the solutions to B and plug into A:  EMBED Equation.DSMT4 . We see that TA =  EMBED Equation.DSMT4 . Do you see how and where we got our extraneous (extra) solution? So how do we avoid this problem. Squaring both sides of an equation will most often not result in an equivalent equation. Check the superset (TB). With other methods we really dont need to check our answers if were sure weve maintained equivalence (double arrows!). Sometimes another solution method wont give us this problem. Heres an easy but instructive example. Ex/ Solve:  EMBED Equation.DSMT4  Method 1 leads to a common error. Method 2 is much safer. Method 1/ Take the square root of both sides Method 2/ Set equal to zero  EMBED Equation.DSMT4   EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (nice algebra) vs  EMBED Equation.DSMT4 (oops!)  EMBED Equation.DSMT4  (factor)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Solving irrational equations (continued) With our graphing/solver calculators we can find decimal answers easily. What well do here is learn a few weird tricks and practice our algebra. Ex/ (A) EMBED Equation.DSMT4  (With 2 radicals things can get messy.) (standard) Method 1/ Square both sides Method 2/ Multiply by the radical conjugate First isolate one radical on one side.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Square both sides. difference of 2 squares (a-b)(a+b) = a2 b2  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Isolate the remaining radical on one side. Combining like terms and rewriting  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Adding to (A) Combine like terms and set equal to zero.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Cross-multiply and square both sides. Hmm, we still have to factor this!  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  x = 4, 84 (but we still have to check them)  EMBED Equation.DSMT4  EMBED Equation.DSMT4  x = 4, 84 Either way,  EMBED Equation.DSMT4  Ex/ (A)  EMBED Equation.DSMT4  Whoa! (Keanu Reeves imitation!) 2 cube roots! Method 1/ u and v substitution?!? Method 2/ Binomial expansion?!? Let  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  (a+b)3 = a3 + 3a2b + 3ab2 + b3 Then we have: u + v = 4 The plan here is to cube both sides of (A). and u3 + v3 = 28 (= 15+2x+13-2x) where (A) is seen as a + b = 4 Factoring a sum of 2 cubes & dividing  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  We get:  EMBED Equation.DSMT4  ab(a+b) = 12 and (A):  EMBED Equation.DSMT4  and (A): a+b = 4 (So divide or replace) Solving this system (let v = 4-u) ab = 3  EMBED Equation.DSMT4  Now cube both sides.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (u 1)(u 3) = 0 195 4x 4x2 = 27 EMBED Equation.DSMT4   EMBED Equation.DSMT4  (x + 7)(x 6) = 0  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  G. Exponents and Exponential Functions 4. Equation Solving (continued) Solving exponential equations involves the unknown quantity (usually x) in the exponent position with various constants for bases. Ex/ The easiest exponential equation to solve is when the bases are the same. 3x = 34 iff x = 4 (Since y = 3x is a 1-1 function. See its graph above.) Ex/ Another easy exponential equation to solve involves base 10. 10x = 7 (It turns out that the log function, y = log10 x, is the inverse function of exponential function, y = 10x. So all we have to do is take the log of both sides of the equation.)  EMBED Equation.DSMT4 log10 10x = log10 7 (log10 10x is the same as f-1(f(x)) which equals x.)  EMBED Equation.DSMT4  x =  EMBED Equation.DSMT4  also written log 7 ,called the common log when base is 10. Ex/ Another easy change of base 10x = 100  EMBED Equation.DSMT4 10x = 102  EMBED Equation.DSMT4  x = 2 Ex/ This one looks tricky  EMBED Equation.DSMT4   EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4 2x2 + 8x 9 = 1  EMBED Equation.DSMT4  2x2 + 8x 10 = 0  EMBED Equation.DSMT4 x2 + 4x 5 = 0  EMBED Equation.DSMT4 (x-1)(x+5) = 0  EMBED Equation.DSMT4 x = 1, -5 Ex/ Heres a fun one.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (I know 8 = 23, but trust me leave it alone!)  EMBED Equation.DSMT4 (Do you see where we headed?) If we make the u-substitution: u = 2x , well get a quadratic in 2x.  EMBED Equation.DSMT4  Now 2x cannot equal -2. So that leaves 2x = 4. Answer:  EMBED Equation.DSMT4  Knowing that function values are equal may not pinpoint the domain value or solution to an equation if the function is not one-to-one. Ex/ Suppose f(x) = x2. Now what if x2 = 32? Does that mean x = 3? (Answer: No!) We did this problem earlier. Since y = x2 is not one-to-one, there may be more than one solution here. Here we know that x = -3, 3 is correct. Timeout! When we write:  EMBED Equation.DSMT4 , were saying, Solutions of the 1st equation are also solutions of the 2nd equation. In terms of truth sets, T1st Equation  EMBED Equation.DSMT4 T2nd Equation. When we write:  EMBED Equation.DSMT4 , were saying, Solutions for the 1st equation solve the 2nd and vice versa. In terms of truth sets, T1st Equation = T2nd Equation. &uv~   : ; G p q    ˸˧˧˧˧ːç˧~zzzzh$ hSH*jh|NhSEHUjY@ hSUV h9hS h9H*h9jh|NhSEHUjX@ hSUVjhSUhSjh|NhBEHUjpX@ hBUVjhBU hBH*hBhB>*hB0'A; b " % W  L v `gdBgd$^gd$`gd$`gdS`gd9c78 ? @ P Q a b f ! 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Exponentiation does  not distribute over addition.ex/3 2 ++4 2 `"3++4() 2 FMathType 4.0 Equation MathType EFEquation.DSMT49q@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  `"1CompObj#%iiObjInfo&kEquation Native l_10881428081)FY/),Y/),Ole pCompObj(*qiObjInfo+sEquation Native t- FMathType 4.0 Equation MathType EFEquation.DSMT49q@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   a  12 a  16 ==a  13 == a 3   with the restriction that a>>0.  FMathType 4.0 Equation MathType EFEquation.DSMT49q_1088142414O.FY/),Y/),Ole }CompObj-/~iObjInfo0@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  a<<0 FMathType 4.0 Equation MathType EFEquation.DSMT49qEquation Native _10881427603FY/),Y/),Ole CompObj24iObjInfo5Equation Native $_10881206288FY/),Y/),Ole @6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  a  16 == a 6 FMathType 4.0 Equation MathTyCompObj79iObjInfo:Equation Native %_10881207366E=FY/),Y/),pe EFEquation.DSMT49q @6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   a  " b  == ab Ole CompObj<>iObjInfo?Equation Native = FMathType 4.0 Equation MathType EFEquation.DSMT49q!@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   "-4  " "-9  == 36  ==6_1088120816BFY/),Y/),Ole CompObjACiObjInfoD FMathType 4.0 Equation MathType EFEquation.DSMT49qu@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   "-4  " Equation Native _1088120982@JGFY/),Y/),Ole CompObjFHi"-9  ==2i"3i==6i 2 ==6("-1)==%"-6 FMathType 4.0 Equation MathType EFEquation.DSMT49q[@6M6GXDSMT4WinAllBasicCodePagesObjInfoIEquation Native w_1088121508LFY/),Y/),Ole Times New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  ab()   12 ==a   12 b   12 FMathType 4.0 Equation MathType EFEquation.DSMT49qCompObjKMiObjInfoNEquation Native d_1088143341'cQFY/),Y/),H@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   a 24 ==a   24 ==a   12Ole CompObjPRiObjInfoSEquation Native E FMathType 4.0 Equation MathType EFEquation.DSMT49q)@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   "-4() 24 == 16 4 ==2`""-4()   12 == "-4  ==%2i (correct and imaginary) FMathType 4.0 Equation MathType EFEquation.DSMT49q_1088146744VFY/),Y/),Ole CompObjUWiObjInfoXc@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  a  1n  means the nth root of aCEquation Native _1088144354^[F 0), 0),Ole CompObjZ\i FMathType 4.0 Equation MathType EFEquation.DSMT49qG@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  a  mn a" a mn  ObjInfo]Equation Native c_1088144424`Ff2),f2),Ole for a>>0 and m,n"!(natural numbers or positive integers) FMathType 4.0 Equation MathType EFEquation.DSMT49qCompObj_aiObjInfobEquation Native _1088144519YeF 0), 0),@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  a>>0 FMathType 4.0 Equation MathType EFEquation.DSMT49qOle CompObjdfiObjInfogEquation Native e@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  a  mn a" a mn == a n () m_1088144751,jF 0), 0),Ole CompObjikiObjInfol FMathType 4.0 Equation MathType EFEquation.DSMT49q@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A  a  mnEquation Native 4_1088144784oF 0), 0),Ole CompObjnpi  #$%&'(),/01234569<=>?@ABEHIJKLMPSTUVWXYZ]`abcfijklmpstuvwz}~ a" a mn FMathType 4.0 Equation MathType EFEquation.DSMT49q@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_ObjInfoqEquation Native _1088144826mwtF 0), 0),Ole  A  a<<0 FMathType 4.0 Equation MathType EFEquation.DSMT49qG@6M6GXDSMT4WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_CompObjsu iObjInfov Equation Native  c_1088144935yF 0), 0),A  for m,n"!(natural numbers) FMathType 4.0 Equation MathType EFEquation.DSMT49q@6M6GXDSMT4WinAllBasicCodePagesOle CompObjxziObjInfo{Equation Native Times New RomanSymbolCourier NewMT Extra!O'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   mn is in 'lowest terms' (ie, reduced). 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