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Exponential and Logarithmic Functions Chapter 4 Composite Functions Section 4.1 Composite Functions Construct new function from two given functions f and g Composite function: Denoted by f ° g Read as “f composed with g” Defined by (f ° g)(x) = f(g(x)) Domain: The set of all numbers x in the domain of g such that g(x) is in the domain of f. Composite Functions Note that we perform the inside function g(x) first. Composite Functions Composite Functions Example. Suppose that f(x) = x3 { 2 and g(x) = 2x2 + 1. Find the values of the following expressions. (a) Problem: (f ± g)(1) Answer: (b) Problem: (g ± f)(1) Answer: (c) Problem: (f ± f)(0) Answer: Composite Functions Example. Suppose that f(x) = 2x2 + 3 and g(x) = 4x3 + 1. (a) Problem: Find f ± g. Answer: (b) Problem: Find the domain of f ± g. Answer: (c) Problem: Find g ± f. Answer: (d) Problem: Find the domain of f ± g. Answer: Composite Functions Example. Suppose that f(x) = g(x) = and (a) Problem: Find f ± g. Answer: (b) Problem: Find the domain of f ± g. Answer: (c) Problem: Find g ± f. Answer: (d) Problem: Find the domain of f ± g. Answer: Composite Functions Example. Problem: If f(x) = 4x + 2 and g(x) = show that for all x, (f ± g)(x) = (g ± f)(x) = x Decomposing Composite Functions Example. Problem: Find functions f and g such that f ± g = H if Answer: Key Points Composite Functions Decomposing Composite Functions One-to-One Functions; Inverse Functions Section 4.2 One-to-One Functions One-to-one function: Any two different inputs in the domain correspond to two different outputs in the range. If x1 and x2 are two different inputs of a function f, then f(x1) f(x2). One-to-One Functions One-to-one function Not a one-to-one function Not a function One-to-One Functions Example. Problem: Is this function one-to-one? Answer: Person Melissa John Jennifer Patrick Salary $45,000 $40,000 $50,000 One-to-One Functions Example. Problem: Is this function one-to-one? Answer: Person Alex Kim Dana Pat ID Number 1451678 1672969 2004783 1914935 One-to-One Functions Example. Determine whether the following functions are one-to-one. (a) Problem: f(x) = x2 + 2 Answer: (b) Problem: g(x) = x3 { 5 Answer: One-to-One Functions Theorem. A function that is increasing on an interval I is a one-to-one function on I. A function that is decreasing on an interval I is a one-to-one function on I. Horizontal-line Test If every horizontal line intersects the graph of a function f in at most one point, then f is one-to-one. Horizontal-line Test Example. Problem: Use the graph to determine whether the function is one-to-one. Answer: 6 4 2 -6 -4 -2 2 -2 -4 -6 4 6 Horizontal-line Test Example. Problem: Use the graph to determine whether the function is one-to-one. Answer: 6 4 2 -6 -4 -2 2 -2 -4 -6 4 6 Inverse Functions Requires f to be a one-to-one function The inverse function of f Written f{1 Defined as the function which takes f(x) as input Returns the output x. In other words, f{1 undoes the action of f f{1(f(x)) = x for all x in the domain of f f(f{1(x)) = x for all x in the domain of f{1 Inverse Functions Example. Find the inverse of the function shown. Problem: Person Alex Kim Dana Pat ID Number 1451678 1672969 2004783 1914935 Inverse Functions Example. (cont.) Answer: ID Number 1451678 1672969 2004783 1914935 Person Alex Kim Dana Pat Inverse Functions Example. Problem: Find the inverse of the function shown. f(0, 0), (1, 1), (2, 4), (3, 9), (4, 16)g Answer: Domain and Range of Inverse Functions If f is one-to-one, its inverse is a function. The domain of f{1 is the range of f. The range of f{1 is the domain of f Domain and Range of Inverse Functions Example. Problem: Verify that the inverse of f(x) = 3x { 1 is Graphs of Inverse Functions The graph of a function f and its inverse f{1 are symmetric with respect to the line y = x. Graphs of Inverse Functions Example. Problem: Find the graph of the inverse function Answer: 6 4 2 -6 -4 -2 2 -2 -4 -6 4 6 Finding Inverse Functions If y = f(x), Inverse if given implicitly by x = f(y). Solve for y if possible to get y = f {1(x) Process Step 1: Interchange x and y to obtain an equation x = f(y) Step 2: If possible, solve for y in terms of x. Step 3: Check the result. Finding Inverse Functions Example. Problem: Find the inverse of the function Answer: Restricting the Domain If a function is not one-to-one, we can often restrict its domain so that the new function is one-to-one. Restricting the Domain Example. Problem: Find the inverse of if the domain of f is x ¸ 0. Answer: 6 4 2 -6 -4 -2 2 -2 -4 -6 4 6 Key Points One-to-One Functions Horizontal-line Test Inverse Functions Domain and Range of Inverse Functions Graphs of Inverse Functions Finding Inverse Functions Restricting the Domain Exponential Functions Section 4.3 Exponents For negative exponents: For fractional exponents: Exponents Example. Problem: Approximate 3¼ to five decimal places. Answer: Laws of Exponents Theorem. [Laws of Exponents] If s, t, a and b are real numbers with a > 0 and b > 0, then as ¢ at = as+t (as)t = ast (ab)s = as ¢ bs 1s = 1 a0 = 1 Exponential Functions Exponential function: function of the form f(x) = ax where a is a positive real number (a > 0) a 1. Domain of f: Set of all real numbers. Warning! This is not the same as a power function. (A function of the form f(x) = xn) Exponential Functions Theorem. For an exponential function f(x) = ax, a > 0, a 1, if x is any real number, then Graphing Exponential Functions Example. Problem: Graph f(x) = 3x Answer: 6 4 2 -6 -4 -2 2 -2 -4 -6 4 6 Graphing Exponential Functions Properties of the Exponential Function Properties of f(x) = ax, a > 1 Domain: All real numbers Range: Positive real numbers; (0, 1) Intercepts: No x-intercepts y-intercept of y = 1 x-axis is horizontal asymptote as x {1 Increasing and one-to-one. Smooth and continuous Contains points (0,1), (1, a) and Properties of the Exponential Function f(x) = ax, a > 1 Properties of the Exponential Function Properties of f(x) = ax, 0 < a < 1 Domain: All real numbers Range: Positive real numbers; (0, 1) Intercepts: No x-intercepts y-intercept of y = 1 x-axis is horizontal asymptote as x 1 Decreasing and one-to-one. Smooth and continuous Contains points (0,1), (1, a) and Properties of the Exponential Function f(x) = ax, 0 < a < 1 The Number e Number e: the number that the expression approaches as n 1. Use ex or exp(x) on your calculator. The Number e Estimating value of e n = 1: 2 n = 2: 2.25 n = 5: 2.488 32 n = 10: 2.593 742 460 1 n = 100: 2.704 813 829 42 n = 1000: 2.716 923 932 24 n = 1,000,000,000: 2.718 281 827 10 n = 1,000,000,000,000: 2.718 281 828 46 Exponential Equations If au = av, then u = v Another way of saying that the function f(x) = ax is one-to-one. Examples. (a) Problem: Solve 23x {1 = 32 Answer: (b) Problem: Solve Answer: Key Points Exponents Laws of Exponents Exponential Functions Graphing Exponential Functions Properties of the Exponential Function The Number e Exponential Equations Logarithmic Functions Section 4.4 Logarithmic Functions Logarithmic function to the base a a > 0 and a 1 Denoted by y = logax Read “logarithm to the base a of x” or “base a logarithm of x” Defined: y = logax if and only if x = ay Inverse function of y = ax Domain: All positive numbers (0,1) Logarithmic Functions Examples. Evaluate the following logarithms (a) Problem: log7 49 Answer: (b) Problem: Answer: (c) Problem: Answer: Logarithmic Functions Examples. Change each exponential expression to an equivalent expression involving a logarithm (a) Problem: 2¼ = s Answer: (b) Problem: ed = 13 Answer: (c) Problem: a5 = 33 Answer: Logarithmic Functions Examples. Change each logarithmic expression to an equivalent expression involving an exponent. (a) Problem: loga 10 = 7 Answer: (b) Problem: loge t = 4 Answer: (c) Problem: log5 17 = z Answer: Domain and Range of Logarithmic Functions Logarithmic function is inverse of the exponential function. Domain of the logarithmic function Same as range of the exponential function All positive real numbers, (0, 1) Range of the logarithmic function Same as domain of the exponential function All real numbers, ({1, 1) Domain and Range of Logarithmic Functions Examples. Find the domain of each function (a) Problem: f(x) = log9(4 { x2) Answer: (b) Problem: Answer: Graphing Logarithmic Functions Example. Graph the function Problem: f(x) = log3 x 6 Answer: 4 2 -6 -4 -2 2 -2 -4 -6 4 6 Properties of the Logarithmic Function Properties of f(x) = loga x, a > 1 Domain: Positive real numbers; (0, 1) Range: All real numbers Intercepts: x-intercept of x = 1 No y-intercepts y-axis is horizontal asymptote Increasing and one-to-one. Smooth and continuous Contains points (1,0), (a, 1) and Properties of the Logarithmic Function Properties of the Logarithmic Function Properties of f(x) = loga x, 0 < a < 1 Domain: Positive real numbers; (0, 1) Range: All real numbers Intercepts: x-intercept of x = 1 No y-intercepts y-axis is horizontal asymptote Decreasing and one-to-one. Smooth and continuous Contains points (1,0), (a, 1) and Properties of the Logarithmic Function Special Logarithm Functions Natural logarithm: y = ln x if and only if x = ey ln x = loge x Common logarithm: y = log x if and only if x = 10y log x = log10 x Special Logarithm Functions Example. Graph the function Problem: f(x) = ln (3{x) Answer: 6 4 2 -6 -4 -2 2 -2 -4 -6 4 6 Logarithmic Equations Examples. Solve the logarithmic equations. Give exact answers. (a) Problem: log4 x = 3 Answer: (b) Problem: log6(x{4) = 3 Answer: (c) Problem: 2 + 4 ln x = 10 Answer: Logarithmic Equations Examples. Solve the exponential equations using logarithms. Give exact answers. (a) Problem: 31+2x= 243 Answer: (b) Problem: ex+8 = 3 Answer: Key Points Logarithmic Functions Domain and Range of Logarithmic Functions Graphing Logarithmic Functions Properties of the Logarithmic Function Special Logarithm Functions Logarithmic Equations Properties of Logarithms Section 4.5 Properties of Logarithms Theorem. [Properties of Logarithms] For a > 0, a 1, and r some real number: loga 1 = 0 loga a = 1 loga ar = r Properties of Logarithms Theorem. [Properties of Logarithms] For M, N, a > 0, a 1, and r some real number: loga (MN) = loga M + loga N loga Mr = r loga M Properties of Logarithms Examples. Evaluate the following expressions. (a) Problem: Answer: (b) Problem: log140 10 + log140 14 Answer: (c) Problem: 2 ln e2.42 Answer: Properties of Logarithms Examples. Evaluate the following expressions if logb A = 5 and logbB = {4. (a) Problem: logb AB Answer: (b) Problem: Answer: (c) Problem: Answer: Properties of Logarithms Example. Write the following expression as a sum of logarithms. Express all powers as factors. Problem: Answer: Properties of Logarithms Example. Write the following expression as a single logarithm. Problem: loga q { loga r + 6 loga p Answer: Properties of Logarithms Theorem. [Properties of Logarithms] For M, N, a > 0, a 1, If M = N, then loga M = loga N If loga M = loga N, then M = N Comes from fact that exponential and logarithmic functions are inverses. Logarithms with Bases Other than e and 10 Example. Problem: Approximate log3 19 rounded to four decimal places Answer: Logarithms with Bases Other than e and 10 Theorem. [Change-of-Base Formula] If a 1, b 1 and M are all positive real numbers, then In particular, Logarithms with Bases Other than e and 10 Examples. Approximate the following logarithms to four decimal places (a) Problem: log6.32 65.16 Answer: (b) Problem: Answer: Key Points Properties of Logarithms Properties of Logarithms Logarithms with Bases Other than e and 10 Logarithmic and Exponential Equations Section 4.6 Solving Logarithmic Equations Example. Problem: Solve log3 4 = 2 log3 x algebraically. Answer: Solving Logarithmic Equations Example. Problem: Solve log3 4 = 2 log3 x graphically. Answer: Solving Logarithmic Equations Example. Problem: Solve log2(x+2) + log2(1{x) = 1 algebraically. Answer: Solving Logarithmic Equations Example. Problem: Solve log2(x+2) + log2(1{x) = 1 graphically. Answer: Solving Exponential Equations Example. Problem: Solve 9x { 3x { 6 = 0 algebraically. Answer: Solving Exponential Equations Example. Problem: Solve 9x { 3x { 6 = 0 graphically. Answer: Solving Exponential Equations Example. Problem: Solve 3x = 7 algebraically. Give an exact answer, then approximate your answer to four decimal places. Answer: Solving Exponential Equations Example. Problem: Solve 3x = 7 graphically. Approximate your answer to four decimal places. Answer: Solving Exponential Equations Example. Problem: Solve 5 ¢ 2x = 3 algebraically. Give an exact answer, then approximate your answer to four decimal places. Answer: Solving Exponential Equations Example. Problem: Solve 5 ¢ 2x = 3 graphically. Approximate your answer to four decimal places. Answer: Solving Exponential Equations Example. Problem: Solve 2x{1 = 52x+3 algebraically. Give an exact answer, then approximate your answer to four decimal places. Answer: Solving Exponential Equations Example. Problem: Solve e2x { x2 = 3 graphically. Approximate your answer to four decimal places. Answer: Key Points Solving Logarithmic Equations Solving Exponential Equations Compound Interest Section 4.7 Simple Interest Simple Interest Formula Principal of P dollars borrowed for t years at per annum interest rate r Interest is I = Prt r must be expressed as decimal Compound Interest Payment period Annually: Once per year Semiannually: Twice per year Quarterly: Four times per year Monthly: 12 times per year Daily: 365 times per year Compound Interest Theorem. [Compound Interest Formula] The amount A after t years due to a principal P invested at an annual interest rate r compounded n times per year is Compound Interest Example. Find the amount that results from the investment of $1000 at 8% after a period of 8 years. (a) Problem: Compounded annually Answer: (b) Problem: Compounded quarterly Answer: (c) Problem: Compounded daily Answer: Compound Interest Theorem. [Continuous Compounding] The amount A after t years due to a principal P invested at an annual interest rate r compounded continuously is Compound Interest Example. Find the amount that results from the investment of $1000 at 8% after a period of 8 years. Problem: Compounded continuously Answer: Effective Rates of Interest Effective Rate of Interest: Equivalent annual simple interest rate that yields same amount as compounding after 1 year. Effective Rates of Interest Example. Find the effective rate of interest on an investment at 8% (a) Problem: Compounded monthly Answer: (a) Problem: Compounded daily Answer: (a) Problem: Compounded continuously Answer: Present Value Present value: amount needed to invest now to receive A dollars at a specified future time. Present Value Theorem. [Present Value Formulas] The present value P of A dollars to be received after t years, assuming a per annum interest rate r compounded n times per year, is if the interest is compounded continuously, then Present Value Example. Problem: Find the present value of $5600 after 4 years at 10% compounded semiannually. Round to the nearest cent. Answer: Time to Double an Investment Example. Problem: What annual rate of interest is required to double an investment in 8 years? Answer: Key Points Simple Interest Compound Interest Effective Rates of Interest Present Value Time to Double an Investment Exponential Growth and Decay; Newton’s Law; Logistic Growth and Decay Section 4.8 Uninhibited Growth and Decay Uninhibited Growth: No restriction to growth Examples Cell division (early in process) Compound Interest Uninhibited Decay Examples Radioactive decay Compute half-life Uninhibited Growth and Decay Uninhibited Growth: N(t) = N0 ekt, k > 0 N0: initial population k: positive constant t: time Uninhibited Decay A(t) = A0 ekt, k < 0 N0: initial amount k: negative constant t: time Uninhibited Growth and Decay Example. Problem: The size P of a small herbivore population at time t (in years) obeys the function P(t) = 600e0.24t if they have enough food and the predator population stays constant. After how many years will the population reach 1800? Answer: Uninhibited Growth and Decay Example. Problem: The half-life of carbon 14 is 5600 years. A fossilized leaf contains 12% of its normal amount of carbon 14. How old is the fossil (to the nearest year)? Answer: Newton’s Law of Cooling Temperature of a heated object decreases exponentially toward temperature of surrounding medium Newton’s Law of Cooling The temperature u of a heated object at a given time t can be modeled by u(t) = T + (u0 { T)ekt, k < 0 where T is the constant temperature of the surrounding medium, u0 is the initial temperature of the heated object, and k is a negative constant. Newton’s Law of Cooling Example. Problem: The temperature of a dead body that has been cooling in a room set at 70±F is measured as 88±F. One hour later, the body temperature is 87.5±F. How long (to the nearest hour) before the first measurement was the time of death, assuming that body temperature at the time of death was 98.6±F? Answer: Logistic Model Uninhibited growth is limited in actuality Growth starts off like exponential, then levels off This is logistic growth Population approaches carrying capacity Logistic Model Logistic Model In a logistic growth model, the population P after time t obeys the equation where a, b and c are constants with c > 0 (c is the carrying capacity). The model is a growth model if b > 0; the model is a decay model if b < 0. Logistic Model Logistic Model Properties of Logistic Function Domain is set of all real numbers Range is interval (0, c) Intercepts: no x-intercept y-intercept is P(0). Increasing if b > 0, decreasing if b < 0 Inflection point when P(t) = 0.5c Graph is smooth and continuous Logistic Model Example. The logistic growth model represents the population of a species introduced into a new territory after t years. (a) Problem: What was the initial population introduced? Answer: (b) Problem: When will the population reach 80? Answer: (c) Problem: What is the carrying capacity? Answer: Key Points Uninhibited Growth and Decay Newton’s Law of Cooling Logistic Model Building Exponential, Logarithmic, and Logistic Models from Data Section 4.9 Fitting an Exponential Function to Data Example. The population (in hundred thousands) for the Colonial US in tenyear increments for the years 1700-1780 is given in the following table. (Source: 1998 Information Please Almanac) Decade, x Population, P 0 251 1 332 2 466 3 629 4 906 5 1171 6 1594 7 2148 8 2780 Fitting an Exponential Function to Data Example. (cont.) (a) Problem: State whether the data can be more accurately modeled using an exponential or logarithmic function. Answer: Fitting an Exponential Function to Data Example. (cont.) (b) Problem: Find a model for population (in hundred thousands) as a function of decades since 1700. Answer: Fitting a Logarithmic Function to Data Example. The death rate (in deaths per 100,000 population) for 2024 year olds in the US between 19851993 are given in the following table. (Source: NCHS Data Warehouse) Year Rate of Death, r 1985 134.9 1987 154.7 1989 162.9 1991 174.5 1992 182.2 Fitting a Logarithmic Function to Data Example. (cont.) (a) Problem: Find a model for death rate in terms of x, where x denotes the number of years since 1980. Answer: (b) Problem: Predict the year in which the death rate first exceeded 200. Answer: Fitting a Logistic Function to Data Example. A mechanic is testing the cooling system of a boat engine. He measures the engine’s temperature over time. Time t (min.) Temperature T (±F) 5 100 10 180 15 270 20 300 25 305 Fitting a Logistic Function to Data Example. (cont.) (a) Problem: Find a model for the temperature T in terms of t, time in minutes. Answer: (b) Problem: What does the model imply will happen to the temperature as time passes? Answer: Key Points Fitting an Exponential Function to Data Fitting a Logarithmic Function to Data Fitting a Logistic Function to Data