Math 80 Chapter 4 Lecture Notes Professor Miguel Ornelas
M. Ornelas Math 80 Lecture Notes Section 4. Section 4. Inverse Functions Definition of One-to-One Function A function f with domain D and range R is a one-to-one function if either of the following equivalent conditions is satisfied:. Whenever a b in D, then f (a) f (b) in R.. Whenever f (a) = f (b) in R, then a = b in D. Determine whether the function f is one-to-one.. f (x) = x + 5. f (x) = x 5 Horizontal Line Test A function f is one-to-one if and only if every horizontal line intersects the graph of f in at most one point. Use the horizontal line test to determine if the function is one-to-one. y y y 5 5 5 4 4 4-5 -4 - - - 4 5 - x -5-4 - - - 4 5 - x -5-4 - - - 4 5 - x - - - - - - -4-4 -4-5 -5-5 Section 4. continued on next page...
M. Ornelas Math 80 Lecture Notes Section 4. (continued) Theorem on Inverse Functions Let f be a one-to-one function with domain D and range R. Function g is the inverse function of f if and only if both of the following conditions are true:. g( f (x)) = x for every x in D. f (g(y)) = y for every y in R Use the theorem on inverse functions to prove that f (x) = x +5, x 0 and g(x) = x 5, x 5 are inverse functions of each other. Domain and Range of f and f. domain of f = range of f. range of f = domain of f Determine the domain and range of f for f (x) = 5 x + without finding f. Section 4. continued on next page...
M. Ornelas Math 80 Lecture Notes Section 4. (continued) Finding the Inverse of a One-to-One Function. Replace f (x) with y. Interchange x and y. Solve the equation for y 4. Replace y with the notation f (x) Find the inverse function of f.. f (x) = 7 x. f (x) = x + x 5 Let f (x) = x. Find the inverse function f and sketch the graphs of f and f on the same coordinate plane. y x Section 4. continued on next page... 4
M. Ornelas Math 80 Lecture Notes Section 4. (continued) Section 4. Exponential Functions Solve the equation.. 6 7 x = 6 x+. 7 x = 9 x Sketch the graphs of f (x) = ( ) x and g(x) = ( ) x on the same coordinate plane. y x f (x) = ( ) x 0 - - - x g(x) = ( ) x 0 - - - 9 8 7 6 5 4-5 -4 - - - 4 5 - x Sketch the graph of f (x) = x+. y 9 x f (x) = x+ -4 - - - 0 8 7 6 5 4-5 -4 - - - 4 5 - x Section 4. continued on next page... 5
M. Ornelas Math 80 Lecture Notes Section 4. (continued) Find an exponential function of the form f (x) = ba x that has y-intercept 8 and passes through point P(, ). A drug is eliminated from the body through urine. Suppose that for an initial dose of 0 milligrams, the amount A(t) in the body t hours later is given by A(t) = 0(0.8) t.. Estimate the amount of the drug in the body 8 hours after the initial dose.. What percentage of the drug still in the body is eliminated each hour? Compound Interest Formula If P dollars are deposited in an account with annual interest rate r, compounded n times per year, then the amount of money in the account after t years is given by the formula ( A(t) = P + r ) nt n If a savings fund pays interest at a rate of % per year compounded semiannually, how much money invested now will amount to $5000 after year? Section 4. continued on next page... 6
M. Ornelas Math 80 Lecture Notes Section 4. (continued) Section 4. The Natural Exponential Function The Number e If n is a positive integer, then ( + n) n e.788 as n Continuously Compounded Interest Formula A = Pe rt Law of Growth (or Decay) Formula q(t) = q 0 e rt The 980 population of the US was approximately million, and the population has been growing continuously at a rate of.0% per year. Predict the population N(t) in the year 00 if this growth trend continues. The radioactive iodine isotope I, used in nuclear imaging, decays continuously at a rate of 5.5% per hour.. Approximate the percentage remaining of any initial amount after 6.4 hours.. What is the half-life of I? Section 4. continued on next page... 7
M. Ornelas Math 80 Lecture Notes Section 4. (continued) Section 4.4 Logarithmic Functions Logarithmic Definition If b > 0 and b, then log b x = y means x = b y Write each as an exponential equation. a. log 5 5 = b. log 6 x = 6 c. log 7 x = 5 Write each as a logarithmic equation. a. 4 = 64 b. 6 / = 6 c. 5 = 5 Find the value of each logarithmic expression. a. log 4 6 b. log 0 0 c. log 9 Solve each equation for x. a. log 5 5 = x b. log x 8 = c. log 6 x = d. log = x e. log b = x f. log x 5 = Section 4.4 continued on next page... 8
M. Ornelas Math 80 Lecture Notes Section 4.4 (continued) Simplify. a. log 5 5 b. log 9 9 c. 6 log 6 5 d. log 7 e. log 4 (log 5 5) f. log (log 6 6) Solve the equation. a. log (x + 4) = log ( x) b. log (x 4) = c. log 9 x = d. ln x = ln( x) If interest is compounded continuously at the rate of 4% per year, approximate the number of years it will take an initial deposit of $6000 to grow to $5,000. Section 4.4 continued on next page... 9
M. Ornelas Math 80 Lecture Notes Section 4.4 (continued) Section 4.5 Properties of Logarithms Laws of Logarithms. log b (xy) = log b x + log b y ( ) x. log b = log y b x log b y. log b (x r ) = r log b x Write the expression as a single logarithm. a. log 5 + log x b. log 5 + log c. log 4 + log 4 0 log 4 5 d. log 0 a + log 0 b log 0 c Expand each expression as much as a possible. a. log 7 5x 4 b. log b 7x c. log y z d. log 6 x x + Section 4.5 continued on next page... 0
M. Ornelas Math 80 Lecture Notes Section 4.5 (continued) Solve. a. log 6 (x ) = log 6 4 log 6 b. log (x + 7) + log x = c. log (x ) + log (x 4) = d. log(x + 4) = log(x ) Section 4.6 Properties of Logarithms Change of Base Formula log b u = log a u log a b Estimate log 5 using the change of base formula. Section 4.6 continued on next page...
M. Ornelas Math 80 Lecture Notes Section 4.6 (continued) Find the exact solution, and a two-decimal-place approximation of each solution, when appropriate. a. x = 7 b. x+4 = x Solve the equation without using a calculator. a. log(log x) = b. e x + e x 5 = 0 Use common logarithms to solve for x in terms of y. y = 0x 0 x 0 x + 0 x Section 4.6 continued on next page...