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# Chapter 6: Exponential and Logarithmic Functions MAT 109

Barry University

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This 36 page Bundle was uploaded by Sterling on Wednesday September 14, 2016. The Bundle belongs to MAT 109 at Barry University taught by Dr. Singh in Fall 2016. Since its upload, it has received 8 views. For similar materials see Precalculus Mathematics 1 in Mathmatics at Barry University.

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Date Created: 09/14/16

MAT 109 Precalculus Mathematics 1 6.1 Composite Functions Notes L. Sterling September 6th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Arithmetic Combinations of Functions f and g are both functions with overlapping domains. 1.1 Sum (f + g)(x) = f (x) + g (x) 1.2 Di▯erence (f ▯ g)(x) = f (x) ▯ g (x) 1.3 Product (fg)(x) = f (x) ▯ g (x) 1 1.4 Quotient ▯ ▯ f f (x) g (x) = g (x) g (x6 0 2 Composition of Two Functions f ▯ g = f composed with g (f ▯ g)(x) = f (g (x)) 0 Domain of g : g (x) is in f s domain 3 Compositions 3.1 f of g of x f ▯ g = f (g (x)) 3.2 f of f of x f ▯ f = f (f (x)) 3.3 g of f of x g ▯ f = g (f (x)) 3.4 g of g of x g ▯ g = g (g (x)) 2 4 How are some steps to ▯nding the composi- tion of two functions? 4.1 Rewrite the composition, which can be like f ▯ g = f (g (x)). 4.2 Replace xs occurrence in the outside function with/into the inside function. 4.3 Simplify as much as possible. 3 MAT 109 Precalculus Mathematics 1 6.2 One-to-One Functions; Inverse Functions Notes L. Sterling September 7th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 One-to-One 1.1 Any function, f(x) = y, for having every single range element in a function is corresponding to only one element of the domain. 2 Horizontal Line Test 2.1 A function noting whether it’s one-to-one if the hori- zontal line hits the graph more than once or not. 1 3 Vertical Line Test 3.1 A function noting whether it’s one-to-one if the ver- tical line hits the graph more than once or not. 4 Function Theorem 4.1 A function that’s increasing on an interval, I, is a one▯ to ▯ one function in I. 4.2 A function that’s decreasing on an interval, I, is a one▯ to ▯ one function on I. 5 Inverse Function 5.1 The given correspondence from f’s range back to f’s ▯1 domain, which can also be described as f . 6 Domain and Range f’s Domain=f1’s Range f’ :Range=f1’s Domain ▯1 7 f and f x ! f (x) ! f(f (x)) = x Domain of f : f(f (x)) = x ▯ ▯ x ! f▯1(x) ! f f1(x) = x 2 ▯ ▯ Domain of f▯1: f f▯1(x) = x 8 Graph Theorem ▯1 A function’s graph, f, and, which is its inverse, are fully symmetric, but with respect to the line, which is y = x. 9 What are the steps into ▯nding an inverse function? 9.1 Since y = f(x), interchange both x and y, which are ▯1 the variables, to create x = f(y), which implies f , which would be the inverse function, implicitly. 9.2 Solve the implicit equation for y, which would be in terms of x in order to obtain f ▯1’s explicit form [if possible]. 9.3 Check your result(s) by showing that both f ▯1 (f (x)) = ▯1 x and f (f (x)) = x. 3 MAT 109 Precalculus Mathematics 1 6.3 Exponential Functions Notes L. Sterling September 8th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Law of Exponents 1.1 Negative Exponent 1 a▯m = m; a6=0 a 1.2 Product of Powers m n m+n a ▯ a = a 1.3 Power of a Power (a ) = am▯n= amn 1 1.4 Power of a Product m m m (ab) = a b 1.5 Power of a Monomial m n p m▯p n▯p mp np (a b ) = a b = a b 1.6 Quotient of Powers am m▯n n = a a 1.7 Zero Exponent 0 a = 1 2 Exponential Function 2.1 Formula f (x) = Ca 2.2 a a : Base a : Positive Real Number (a > 0) a 6= 1 C 6= 0 2 2.3 Domain of f All Real Numbers 2.4 Growth Factor a : Growth Factor f (0) = Ca = C (1) = C 2.5 Initial Value C : Initial V alue 3 Exponential Function Theorem 3.1 Function f (x) = C ▯ a a > 0 a 6= 1 3.2 If x : Any Real Number 3.3 Then f (x + 1) f (x) = a f (x + 1) = af (x) 3 4 Exponential Functions Properties 4.1 Function x f (x) = a a > 1 4.2 Domain All Real Numbers 4.3 Range All Positive Real Numbers 4.4 X-Intercept None 4.5 Y-Intercept y = 1 4.6 Horizontal Asymptote X ▯ axis : x ! 1 4.7 Increasing One ▯ to ▯ One 4.8 Points (0; 1) 4 (1; a) ▯ ▯ ▯1; 1 a 4.9 Graph Smooth Continuous No Corners No Gaps 5 Exponential Functions Properties 5.1 Function f (x) = a 0 < a < 1 5.2 Domain All Real Numbers 5.3 Range All Positive Real Numbers 5.4 X-Intercept None 5 5.5 Y-Intercept y = 1 5.6 Horizontal Asymptote X ▯ axis : x ! 1 5.7 Decreasing One ▯ to ▯ One 5.8 Points (0; 1) (1; a) ▯ ▯ ▯1; 1 a 5.9 Graph Smooth Continuous No Corners No Gaps 6 e 6.1 Expression ▯ ▯n 1 + 1 : n ! 1 n 6 6.2 Limit Notation ▯ ▯n lim 1 + 1 = e n!1 n 7 Solving Exponential Equations 7.1 If...then... If : a = av Then : u = v 7 MAT 109 Precalculus Mathematics 1 6.4 Logarithmic Functions Notes L. Sterling September 9th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Logarithmic Function 1.1 Base This would be towards base a with the following: a > 0 a 6= 1 1.2 y = loa x Read as the following: y is the logarithm to the base; a; of x 1 1.3 De▯nition y y = loa x ! x = a 1.4 Domain x > 0 2 Domain and Range 2.1 Logarithmic Functions Domain Exponential Function s Range (0; 1) 2.2 Logarithmic Functions Range 0 Exponential Function s Domain (▯1; 1) 3 y = log x a Logarithm : y = log x a y Exponential : x = a Domain : 0 < x < 1 Range : ▯1 < y < 1 2 4 Properties of f (x) = log x a 4.1 Domain All Positive Real Numbers 4.2 Range All Real Numbers 4.3 X-Intercept 1 4.4 Y-Intercept None 4.5 Vertical Asymptote Y ▯ Axis (x = 0) 4.6 Decreasing 0 < a < 1 4.7 Increasing a > 1 3 4.8 Points (1; 0) (a; 1) ▯ ▯ 1 ; ▯1 a 4.9 Graph Smooth Continuous No Corners No Gaps 5 Natural Logarithm Function y = ln(x) ! x = e y y = log (x) = 10(x) ! x = 10 4 MAT 109 Precalculus Mathematics 1 6.5 Properties of Logarithms Notes L. Sterling September 12th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Properties of Logarithms In This Case : y is just a variable and not f (x): 1.1 Inverse Properties aloa x = x x loa a x = loa (a ) = xloga(a) 1 = x 1.2 Power Property y Logarithm : logax = y logax Natural Logarithm : ln a = y ln xa 1.3 Product Property Logarithm : loga(xy) = logax + loa y loga(xy) 6= (loa x)(loa y) Natural Logarithm : ln axy) = ln a + lnay lna(xy) 6= (la x)(la y) 1.4 Product of the Power and the Log logax = y logax a 6= 1 1.5 Quotient Property ▯ ▯ x Logarithm : log a = logax ▯ logay y ▯ ▯ x log x loga 6= a y logay ▯ ▯ Natural Logarithm : ln x = ln x ▯ ln y a y a a 2 ▯ ▯ x lnax lna y 6=ln y a 2 One-to-One Properties 3 IFF x = y x y a = a loa x = loa y 4 Change-of-Base Formula a 6= 1 b 6= 1 x : Positive Real Numbers logbx logax = logba log x loa x =log a logax = ln x ln a 3 MAT 109 Precalculus Mathematics 1 6.6 Logarithmic and Exponential Equations Notes L. Sterling September 13th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 IFF x = y a = a y logax = loa y 2 Inverse Properties aloa = x log a = log (a ) = xlog (a) = x a a a 1 3 loga x > 0 a > 0 a 6= 1 y = f (x) = aog iff : x = a 4 loga: One ▯ to ▯ One M : Positive Real Numbers N : Positive Real Numbers a : Positive Real Numbers a 6= 1 M = N loa M = loa N x 5 a : One ▯ to ▯ One u : Positive Real Numbers 2 v : Positive Real Numbers a > 0 a 6= 1 u = v a = a v 6 Laws of Logs Theorem M > 0 N > 0 a > 0 a 6= 1 r : Any Real Number logaM = r log a log (MN) = log M + log N a a a loa (MN) 6= (logaM)(log a) 3 ▯ ▯ M loa = loa M ▯ lag N N ▯ ▯ M log M loga 6= a N logaN 4 MAT 109 Precalculus Mathematics 1 6.7 Financial Models Notes L. Sterling September 14th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Simple Interest Formula 1.1 Formula I = Prt 1.2 Parts to the Formula I : Interest P : Principal r : (Per Annum)Interest Rate t : Y ears 1 2 Compounded Interests 2.1 Annually Once per Y ear 2.2 Semiannually Twice per Y ear 2.3 Quarterly 4 Times per Y ear 2.4 Monthly 12 Times per Y ear 2.5 Weekly 52 Times per Y ear 2.6 Daily 365 Times per Y ear 2.7 Not Taught: Daily in a Leap Year 366 Times per Y ear 3 Compounded Interest: Daily ▯ 365 Times per Year 2 ▯ Banks de▯ne interest with the following information: { 1 Month = 30 Days { 1 Year = 360 Days 4 Deriving a Formula for Compound Interest A = P + i A = P + Prt A = P (1 + rt) 4.1 Cycles 1st Cycle : A = P (1 + rt) 1 2 2nd Cycle : 2 = A1(1 + rt) = P (1 + rt)(1 + rt) = P (1 + rt) 3rd Cycle : 3 = A2(1 + rt) = P (1 + rt) (1 + rt) = P (1 + rt) Any Cycle : A = P (1 + rt) k 5 Compound Interest Formula 5.1 Formula ▯ r▯nt A = P a + n 5.2 Parts to the Formula A : Amount P : Principal 3 r : (Per Annum)Interest Rate n : Number of Times per Y ear t : Y ears 6 Compounding 6.1 Annual Compounding n = 1 6.2 Semiannual Compounding n = 2 6.3 Quarterly Compounding n = 4 6.4 Monthly Compounding n = 12 6.5 Weekly Compounding n = 52 6.6 Daily Compounding n = 365 4 6.7 Not Taught: Daily in a Leap Year n = 366 7 Formula Simplifying Note :n = h r ▯ r▯n A = P 1 + n ▯ ▯ n 1 = P 1 + n r "▯ ▯n# r 1 r = P a +n r " ▯ ▯h#r 1 = P 1 +h 8 Continuous Compounding 8.1 Formula A = Pert 8.2 Parts to the Formula A : Amount P : Principal e ▯ 2:718 r : (Per Annum)Interest Rate 5 t : Y ears 9 E▯ective Rate of Interest re: Effective Rate of Interest 9.1 Compounding n Times per Year ▯ ▯ r n e = 1 + n ▯ 1 9.2 Continuous Compounding re= e ▯ 1 10 Present Value Formulas 10.1 Formula: Compounded n Times per Year ▯ r▯▯nt P = A 1 + n 10.2 Parts to the Formula P : Principal A : Amount r : (Per Annum)Interest Rate n : Number of Times per Y ear t : Y ears 6 10.3 Formula: Continuous Compounding P = Ae ▯rt 10.4 Parts to the Formula P : Principal A : Amount e ▯ 2:718 r : (Per Annum)Interest Rate t : Y ears 7 MAT 109 Precalculus Mathematics 1 6.8 Exponential Growth and Decay Models; Newtons Law; Logistic Growth and Decay Models Notes L. Sterling September 15th, 2016 Abstract Provide a generalization to each of the key terms listed in this section. 1 Uninhibited Growth A = A0et 2 Uninhibited Growth of Cells 2.1 Function N (t) = N e ; k > 0 0 1 2.2 Parts of the Function N : Number t : Time N : Initial Number of Cells 0 e ▯ 2:718 k : Positive Constant k > 0 : Any Positive Constant Greater Than 0 3 Uninhibited Radioactive Decay 3.1 Function kt A(t) = A0e ; k < 0 3.2 Parts of the Function A : Radioactive Material t : Time A 0 ORiginal Amount of Radioactive Material e ▯ 2:718 k : Negative Constant k < 0 : Any Negative Constant Less Than 0 2 4 Newtons Law of Cooling 4.1 Function kt u(t) = T + 0u ▯ T)e ; k < 0 4.2 Parts of the Function u : Temperature t : Time T : Constant Temperature u : Initial Temperature 0 e ▯ 2:718 k : Negative Constant k < 0 : Any Negative Constant Less Than 0 5 Logistic Model 5.1 Function P (t) = c 1 + aebt 5.2 Parts of the Function P : Population t : Time c : Constant [c > 0] 3 e ▯ 2:718 a : Constant b : Constant 5.3 Growth and Decay Growth : b > 0 Decay : b < 0 6 Logistic Growth Functions Properties 6.1 Domain All Real Numbers 6.2 Range (0; c) c : Carrying Capacity 6.3 X-Intercepts None 6.4 Y-Intercepts P (0) 6.5 Horizontal Asymptote y = 0 4 y = c 6.6 Increasing and Decreasing Increasing : b > 0 Decreasing : b < 0 6.7 In ection Point ▯ De▯nition: A graphs given point where the graph will change from the following: { For Growth Function: Curving upwards to downwards { For Decay Function: Curving downwards to upwards 1 ▯ In ecting: P (t) 2 [of c] 6.8 Graph Smooth Continuous No Corners No Gaps 5

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